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2015 - 1506(2) - 1507(1) - 1510(1) - 1512(1)

2016 - 1601(2) - 1605(1) - 1608(1) - 1609(1) - 1610(2) - 1611(2) - 1612(1)

2017 - 1701(2) - 1703(1) - 1705(1) - 1706(1) - 1707(2) - 1709(1) - 1710(2) - 1711(2)

Any replacements are listed further down

[67] **viXra:1711.0301 [pdf]**
*submitted on 2017-11-14 11:58:38*

**Authors:** Edigles Guedes

**Comments:** 3 Pages.

In this paper, we construct an equation involving Alexander polynomials for knots 86 and 87.

**Category:** Topology

[66] **viXra:1711.0253 [pdf]**
*submitted on 2017-11-09 00:57:34*

**Authors:** Nicholas R. Wright

**Comments:** 6 Pages.

We find that the Hodge conjecture is an overdetermined system rather than an underdetermined system, thereby becoming homogenous and leading to overfitting. A product of normal spaces need not be normal thereby initiating and reinforcing the conjecture. A norm should be found through regularization. We find the Magic Star polygon to be a satisfying norm. The Torelli theorem also gives deep and technical proof. Analysis can be seen through a multiple regression technique. Automorphism is not obtainable with a contradiction in the trivial and integral.

**Category:** Topology

[65] **viXra:1710.0278 [pdf]**
*submitted on 2017-10-24 06:22:09*

**Authors:** Suehwan Jeong, Junho Yeo

**Comments:** 7 pages, Abstract available on English/Korean both, Main text Korean only

The Four Color Theorem(4CT) is the theorem stating that no more than four colors are required to color each part of a plane divided into finite parts so that no two adjacent parts have the same color. It was proven in 1976 by Kenneth Appel and Wolfgang Haken, but here we will prove 4CT without Computer Resources. We can display the picture to color as a graph, using nodes that are the points that represent each figure in the picture, and stects which is lines that links two nodes neighboring each other. We proved that every picture to be colored can be expressed as a liner graph, made up only with nodes and stects that have the shape of straight lines. We will name the triangle made of nodes and stects that contains other nodes and stect, ‘Triangular Convex Cell’. Also, we will call the graph that has the Triangular Convex Cell and also has the form of a convex set, a ‘Triangular Convex Cell graph’. The Euler characteristic in plane graph is 1, so if we regard the numver of the node v, the number of the stect e, and the number of the sides made by nodes and stects f, the equation v-e+f=1is established. Using this, we can derive the result that the graph with the highest total connection strength, which is the number of the connected graphs of each node in the graph, is a triangular-convex graph. Now, we prove the Triangular Convex Cell graph prove The Four Color Theorem regardless of how many nodes there are in the Triangular Convex Cell, or it’s arranged shape. We can use this to colorize a huge Triangular Convex Cell graph in which there are ∞ nodes inside the Triangular Convex Cell, and delete the nodes and the stects of the graph as needed to fit the picture to be colored. Thus, by using this method, it can be seen that the Four Color Theorem holds for all the pictures.

**Category:** Topology

[64] **viXra:1710.0118 [pdf]**
*submitted on 2017-10-10 23:07:27*

**Authors:** Dmitri Martila

**Comments:** 4 Pages.

It is amazing to see, how the problems find their solutions. Even such extremely long as the 1200 pages of the ABC-hypothesis proof of the ``Japan Perelman'', which is needed to be consumed by the most brilliant men to come. And like the first PCs were huge but became compact, the large proofs can turn into very compact ones. \copyright

**Category:** Topology

[63] **viXra:1709.0381 [pdf]**
*submitted on 2017-09-25 05:22:17*

**Authors:** Choe Ryujin

**Comments:** 5 Pages.

Solution of Inscribed Squares Problem

**Category:** Topology

[62] **viXra:1707.0127 [pdf]**
*submitted on 2017-07-09 11:29:59*

**Authors:** Antoine Balan

**Comments:** 2 pages, written in french

A cohomology is defined with an automorphism of the tangent fiber bundle. The so defined cohomology is a topological invariant of the manifold considered.

**Category:** Topology

[61] **viXra:1707.0065 [pdf]**
*submitted on 2017-07-05 04:20:26*

**Authors:** David Edwards, Robert Varley

**Comments:** 7 Pages.

We define Adelic Homotopy Theory and compare it with Rational Homotopy Theory.

**Category:** Topology

[60] **viXra:1706.0547 [pdf]**
*submitted on 2017-06-28 13:16:36*

**Authors:** Arturo Tozzi, James F Peters, Raquel del Moral, Pedro C Marijuan

**Comments:** 10 Pages.

A unifying principle underlies the organization of physical and biological systems. It relates to a well-known topological theorem which succinctly states that an activity on a planar circumference projects to two activities with “matching description” into a sphere. Here we ask: What does “matching description” mean? Has it something to do with “identity”? Going through different formulations of the principle of identity, we describe diverse possible meanings of the term “matching description”. We demonstrate that the concepts of “sameness”, “equality”, “belonging together” stand for intertwined levels with mutual interactions. By showing that “matching” description is a very general and malleable concept, we provide a novel testable approach to “identity” that yields helpful insights into physical and biological matters. Indeed, we illustrate how a novel mathematical approach derived from the Borsuk-Ulam theorem, termed bio-BUT, might explain the astonishing biological “multiplicity from identity” of evolving living beings as well as the logic of their intricate biochemical arrangements.

**Category:** Topology

[59] **viXra:1703.0035 [pdf]**
*submitted on 2017-03-04 09:18:56*

**Authors:** Antoine Balan

**Comments:** 1 Page. written in French

We define here an homology for a manifold called edge homology.

**Category:** Topology

[58] **viXra:1701.0649 [pdf]**
*submitted on 2017-01-28 01:09:57*

**Authors:** Arturo Tozzi

**Comments:** 3 Pages.

Recently introduced versions of the Borsuk-Ulam theorem (BUT) state that a feature on a n-manifold projects to two features with matching description onto a n+1 manifold. Starting from this rather simple “abstract” claim, a fruitful general framework has been built, able to elucidate disparate “real” physical and biological phenomena, from quantum entanglement, to brain activity, from gauge theories to pre- big bang scenarios. One of the main concerns of such a topological approach to systems features is that it talks in rather general terms, leaving apart the peculiar features of individuals and of single physical and biological processes. In order to tackle this issue, in this brief note we ask: what does it mean “matching description”? has matching description anything to do with “identity”?

**Category:** Topology

[57] **viXra:1701.0519 [pdf]**
*submitted on 2017-01-17 00:26:31*

**Authors:** W. B. Vasantha Kandasamy, K. Ilanthenral, Florentin Smarandache

**Comments:** 278 Pages.

In this book authors for the first time develop the notion of MOD natural neutrosophic subset special type of topological spaces using MOD natural neutrosophic dual numbers or MOD natural neutrosophic finite complex number or MOD natural neutrosophic-neutrosophic numbers and so on to build their respective MOD semigroups. Later they extend this concept to MOD interval subset semigroups and MOD interval neutrosophic subset semigroups. Using these MOD interval semigroups and MOD interval natural neutrosophic subset semigroups special type of subset topological spaces are built. Further using these MOD subsets we build MOD interval subset matrix semigroups and MOD interval subset matrix special type of matrix topological spaces. Likewise using MOD interval natural neutrosophic subsets matrices semigroups we can build MOD interval natural neutrosophic matrix subset special type of topological spaces. We also do build MOD subset coefficient polynomial special type of topological spaces. The final chapter mainly proposes several open conjectures about the validity of the Kakutani’s fixed point theorem for all MOD special type of subset topological spaces.

**Category:** Topology

[56] **viXra:1612.0338 [pdf]**
*submitted on 2016-12-24 17:35:46*

**Authors:** Pablo Álvarez Domínguez

**Comments:** 3 Pages.

The main objectives of this little work is to propose a conjecture about a condition that
every Kissing Number must satisfy and to study a little bit its most basic direct
consequences if it were proven true. It can seem that nowadays there is not enough
acknowledge to conjecture it (mainly because the little information we have about Kissing
Numbers). However, the few known examples we have about this type of numbers satisfy
it.

**Category:** Topology

[55] **viXra:1611.0322 [pdf]**
*submitted on 2016-11-24 03:27:16*

**Authors:** A.a.salama, I.m.hanafy, Hewayda Elghawalby, M.s.dabash

**Comments:** 10 Pages.

In this paper, we introduce and study the concept of "neutrosophic crisp closed set "and
"neutrosophic crisp continuous function. Possible application to GIS topology rules are touched
upon.

**Category:** Topology

[54] **viXra:1611.0223 [pdf]**
*submitted on 2016-11-14 20:12:06*

**Authors:** Vincenzo Nardozza

**Comments:** 13 Pages.

The definition of an essential n-dimensional compact manifold is given. A tensor-like object used as an invariant for compact n-dimensional spaces and a model for describing topological manipulations based on the above invariants is proposed. The proposed theory is not fully proven and therefore this paper rather then being a proper formal paper has to be considered a note that gives some hints for further formal mathematical research.

**Category:** Topology

[53] **viXra:1610.0223 [pdf]**
*submitted on 2016-10-19 03:29:22*

**Authors:** Arturo Tozzi, James F Peters

**Comments:** 8 Pages.

The (spatial) fractals and (temporal) power laws are ubiquitously displayed by large classes of biological systems. Nevertheless, they are controversial phenomena with still unexplained genesis. From the far-flung branch of topology, a helpful concept comes into play, namely the Borsuk-Ulam theorem, shedding new light on the scale-free origin’s long-standing enigma. The theorem states that a single point, if embedded in just one spatial dimension higher, gives rise to two antipodal points that have matching descriptions and similar features. Here we demonstrate that, when we introduce into a system the proper fractal extra-dimension instead of a spatial one, we are able to achieve two antipodal self-similar shapes, corresponding to the distinctive scale-free’s higher and lower magnifications. By showing that the elusive phenomena of fractals and power laws can be explained and analyzed in a topological framework, we make clear why the Borsuk-Ulam theorem is the most general principle underlying their pervasive occurrence in nature.

**Category:** Topology

[52] **viXra:1610.0222 [pdf]**
*submitted on 2016-10-19 03:34:00*

**Authors:** Arturo Tozzi, James F Peters

**Comments:** 8 Pages.

Nonlinear chaotic dynamics are widespread, both in physical and biological systems. This form of dynamics is frequently studied through logistic maps equipped with bifurcations, where intervals are dictated by the Feigenbaum constants. In such a multifaceted framework, a concept from the far-flung branch of topology, namely the Borsuk-Ulam theorem, comes into play. The theorem tells us that a continuous mapping from antipodal points with matching feature values on an n-sphere to the same real value can always be found. Here we demonstrate that embracing nonlinearity in the framework of the Borsuk-Ulam theorem means that bifurcation transformations (the antipodal points) can be described as paths or trajectories on abstract spheres equipped with a Feigenbaum dimension. Such an approach allows the evaluation of nonlinear systems through linear techniques. In conclusion, we provide a general topological mechanism which explains the elusive chaotic phenomena, cast in a physical/biological fashion which has the potential of being operationalized.

**Category:** Topology

[51] **viXra:1609.0084 [pdf]**
*submitted on 2016-09-07 07:48:06*

**Authors:** Mirosław J. Kubiak

**Comments:** 7 Pages.

We proposed a description of the gravitational phenomena in a new medium, which merges the Minkowski four-dimensional spacetime and the bare mass density into the single idea. Under influence outer gravitational field the Minkowski four-dimensional spacetime does not change, while the bare mass density changing and becomes the effective mass density. This is an alternative attempt to describe gravitational phenomena, using a new idea of massification of the spacetime.

**Category:** Topology

[50] **viXra:1608.0003 [pdf]**
*submitted on 2016-08-01 11:41:37*

**Authors:** Richard L Amoroso

**Comments:** 39 Pages.

We thank Newton for inspiring strict adherence to hypotheses non-fingo1,and claim reasonable a posteriori surety in positing the need for an
Ontological-Phase Topological Field Theory (OPTFT) as the final step in
describing the remaining requirements for bulk UQC. Let’s surmise with
little doubt that a radical new theory needs to be correlated with the
looming 3rd regime of Unified Field Mechanics (UFM). If the author
knows one thing for sure, it is that gravity is not quantized! The physics
community is so invested in quantizing the gravitational force that it could
still be years away from this inevitable conclusion. There is still a serious
conundrum to be dealt with however; discovery of the complex Manifold
of Uncertainty (MOU), the associated ‘semi-quantum limit’ and the fact
of a duality between Newton’s and Einstein’s gravity, may allow some
sort of wave-particle-like duality with a quantal-like virtual graviton in the
semi-quantum limit. Why mention the gravitational field? Relativistic
information processing (RIP) introduces gravitational effects in the
‘parallel transport’ aspects of topological switching in branes. There are A
and B type topological string theories, and a related Topological MTheory
with mirror symmetry, that are somewhat interesting especially
since they allow sufficient dimensionality with Calabi-Yau mirror
symmetry perceived as essential elements for developing a UFM. But a
distinction between these theories and the ontology of an energyless
topological switching of information (Shannon related) through
topological charge in brane dynamics, perhaps defined in a manner
making correspondence to a higher dimensional (HD) de-Broglie-Bohm
super-quantum potential synonymous with a 'Force of coherence' of the
unified field is of interest. Thus the term 'OPTFT’ has been chosen to address this issue as best as the Zeitgeist is able to conceive at the time of writing…

**Category:** Topology

[49] **viXra:1605.0237 [pdf]**
*submitted on 2016-05-22 14:15:00*

**Authors:** Luis Sancho

**Comments:** 355 Pages.

The 5th dimension is in the discontinuities of the real line, which grow as we look into smaller scales, till becoming infinite.
The 5th dimension creates different geometries for each different world.
From the human relative frame of reference, the observer has an Euclidean view of its scale, a hyperbolic view of smaller quantum systems and an elliptic view of larger gravitational worlds, according to the ratio between its informative curvature, Tƒ and length of its space quanta Sp.
The symmetries that web bidimensional planes of space with tall, cyclical frequencies of time create the 4D Space-time beings of the fractal Universe, constantly moving, generating, growing, reproducing, evolving new fractal scales of the 5th dimension.
But as time passes the densities of its frequency of time cycles, Tƒ, grow and the system curves in excess, ageing and finally completing a 0-sum closed curve,, ending a world cycle in which the topological transformations of the being, are seen as its 3 ages each one a phase of its r=evolution as a world-view.

**Category:** Topology

[48] **viXra:1601.0106 [pdf]**
*submitted on 2016-01-10 13:54:58*

**Authors:** Alex Patterson

**Comments:** 4 Pages. Paper 1, of program.

Using Functor Substitutability for the basis of creating an intuitionist interpretation of the results of that subscribe. Laying the Grounds for Challenging Goedel's Incompleteness Theorems with the Aspects of the Poincare Group. Proceeding with: Propositional Quantum Logic, Transitioning “Non-Commensurable” Smoothly to Functor-Auxiliary Substitutability and Detachment.

**Category:** Topology

[47] **viXra:1601.0031 [pdf]**
*submitted on 2016-01-05 10:40:24*

**Authors:** H.E. Winkelnkemper

**Comments:** 7 Pages.

Using Artin Presentation Theory, we mathematically augment a
remark of Atiyah on physics and Donaldson's 4D theory which, conversely, explicitly
introduces the theoretical physical relevance of AP Theory into Modern
Physics. AP Theory is a purely discrete group-theoretic, in fact, a framed pure
braid theory, which, in the sharpest possible holographic manner, encodes all
closed, orientable 3-manifolds and their knot and linking theories, and a large
class of compact, connected, simply-connected, smooth 4-manifolds with a connected
boundary, whose physical relevance for Atiyah's remark we explain.

**Category:** Topology

[46] **viXra:1512.0456 [pdf]**
*submitted on 2015-12-28 04:12:15*

**Authors:** Alex Patterson

**Comments:** 19 Pages. Ability to go and move on Against Method, pace PKF spoon listserve archive

A demonstration of using Against Method, of Paul K. Feyerabend.

**Category:** Topology

[45] **viXra:1510.0057 [pdf]**
*submitted on 2015-10-06 03:47:10*

**Authors:** Dmitri Martila

**Comments:** 4 Pages.

There is Prize committee (claymath.org), which requires publication in worldwide reputable mathematics journal and at least two years of following scientific admiration. Why then the God-less Grisha Perelman has published only in a
God-less forum (arXiv), publication was unclear as the crazy sketch; but mummy child "Grisha" has being forced to accept the Millennium Prize? Am I simply ugly or poor? Please respect my copyrights!

**Category:** Topology

[44] **viXra:1507.0204 [pdf]**
*submitted on 2015-07-27 18:50:52*

**Authors:** W. B. Vasantha Kandasamy, Florentin Smarandache

**Comments:** 225 Pages.

In this book for the first time the authors introduce a special type of topological spaces using the interval [0, n). Several of the properties enjoyed by these special spaces are analyzed. Over hundred problems are suggested, some of which are open conjectures. This book gives a new perspective to topological spaces.

**Category:** Topology

[43] **viXra:1506.0184 [pdf]**
*submitted on 2015-06-25 19:16:39*

**Authors:** a. a. Salama

**Comments:** 5 Pages.

Since the world is full of indeterminacy, the neutrosophics
found their place into contemporary research. The fundamental
concepts of neutrosophic set, introduced by Smarandache
in [30, 31, 32] and Salama et al. in [4-29]. In Geographical
information systems (GIS) there is a need to model spatial regions with indeterminate boundary and under indeterminacy. In this paper the structure of some classes of neutrosophic crisp nearly open sets are investigated and some applications are given.Finally we generalize the crisp topological and intuitioistic studies to the notion of neutrosophic crisp set. Possible applications
to GIS topological rules are touched upon.

**Category:** Topology

[42] **viXra:1506.0139 [pdf]**
*submitted on 2015-06-18 12:06:37*

**Authors:** Luis Sancho

**Comments:** 55 Pages.

We expand the holographic principle to explain the nature of the universe and all its fractal 2-manifold organisms.
The simplest explanation of the complex universe departs from the holographic principle: information, form is bidimensional. Because all what we perceive is the 'cover' of things, its membranes.
Thus in a 2-manifold, in a bidimensional world there are only 2 topologies, which correspond to the main parts of any system, the lineal, energetic and spherical informative dimensions or motions (since energy is just space in motion and time information cyclical form in motion). And its 'wavelike combinations, exi.
And we shall call energy, to th expansive motions, which form limbs, often with hyperbolic topology (bilateral).
And we shall call information to the implosive vortices, which form particles and heads, often of spherical geometry.
And the space between them the 3 membrane-topology of the universe, its body wave, the ExI part of the system.
Thus all what exists are polar systems, with 'energy fields/limbs' and informative heads/particles, exchange energy and information creating a 3rd element, 'waves-bodies'.

**Category:** Topology

[41] **viXra:1410.0132 [pdf]**
*submitted on 2014-10-22 17:33:09*

**Authors:** A. A. Salama, Said Broumi, Florentin Smarandache

**Comments:** 8 Pages.

The focus of this paper is to propose a new
notion of neutrosophic crisp sets via neutrosophic crisp
ideals and to study some basic operations and results in
neutrosophic crisp topological spaces. Also,
neutrosophic crisp L-openness and neutrosophic crisp Lcontinuity
are considered as a generalizations for a crisp
and fuzzy concepts. Relationships between the above
new neutrosophic crisp notions and the other relevant
classes are investigated. Finally, we define and study
two different types of neutrosophic crisp functions.
Index Terms—Neutrosophic Crisp Set; Neutrosophic
Crisp Ideals; Neutrosophic Crisp L-open Sets;
Neutrosophic Crisp L- Continuity; Neutrosophic Sets.
I. INTRODUCTION
The fuzzy set was introduced by Zadeh [20] in 1965,
where each element had a degree of membership. In
1983 the intuitionstic fuzzy set was introduced by K.
Atanassov [1, 2, 3] as a generalization of fuzzy set,
where besides the degree of membership and the degree
of non- membership of each element. Salama et al [11]
defined intuitionistic fuzzy ideal and neutrosophic ideal
for a set and generalized the concept of fuzzy ideal
concepts, first initiated by Sarker [19]. Smarandache [16,
17, 18] defined the notion of neutrosophic sets, which is
a generalization of Zadeh's fuzzy set and Atanassov's
intuitionistic fuzzy set. Neutrosophic sets have been
investigated by Salama et al. [4, 5, 6, 7, 8, 9, 10, 11, 12,
13, 14, 15]. In this paper is to introduce and study some
new neutrosophic crisp notions via neutrosophic crisp
ideals. Also, neutrosophic crisp L-openness and
neutrosophic crisp L- continuity are considered.
Relationships between the above new neutrosophic crisp
notions and the other relevant classes are investigated.
Recently, we define and study two different types of
neutrosophic crisp functions.
The paper unfolds as follows. The next section briefly
introduces some definitions related to neutrosophic set
theory and some terminologies of neutrosophic crisp set
and neutrosophic crisp ideal. Section 3 presents
neutrosophic crisp L- open and neutrosophic crisp Lclosed
sets. Section 4 presents neutrosophic crisp L–
continuous functions. Conclusions appear in the last section.

**Category:** Topology

[40] **viXra:1409.0124 [pdf]**
*submitted on 2014-09-16 03:54:14*

**Authors:** Vincenzo Nardozza

**Comments:** 10 Pages.

By means of a computer, all the possible homogeneous compact 3-Delta-complexes made up of a small number of simplexes (from 1 to 3) have been classified in homology classes. The analysis shows that, with a small number of simplexes, it is already possible to build quite a large number of separate topological spaces.

**Category:** Topology

[39] **viXra:1406.0180 [pdf]**
*submitted on 2014-06-30 02:12:30*

**Authors:** S.kalimuthu

**Comments:** 8 Pages. NA

In this work a new topological non linear differential equation has been formulated

**Category:** Topology

[38] **viXra:1406.0153 [pdf]**
*submitted on 2014-06-24 18:39:44*

**Authors:** A. A. Salama, Mohamed Abdelfattah, S. A. Alblowi

**Comments:** 13 Pages.

In Geographical information systems (GIS) there is a need to model spatial regions with intuitionistic
boundary. In this paper, we generalize the topological ideals spaces to the notion of intuitionistic set; we construct the
basic fundamental concepts and properties of an intuitionistic spatial region. In addition, we introduce the notion of
ideals on intuitionistic set which is considered as a generalization of ideals studies in [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14,
15, 16]. The important topological intuitionistic ideal has been given. The concept of intuitionistic local function is also introduced for a intuitionistic topological space. These concepts are discussed with a view to find new intuitionistic topology from the original one. The basic structure, especially a basis for such generated intuitionistic topologies and several relations between different topological intuitionistic ideals are also studied here. Possible application to GIS topology rules are touched upon.

**Category:** Topology

[37] **viXra:1405.0291 [pdf]**
*submitted on 2014-05-22 21:27:19*

**Authors:** Zhipeng Lin

**Comments:** 8 Pages.

Dynamical conserved topology(DCT) has conserved number of nodes and links’ ends, its nodes can exchange with other nodes, its links’ ends
can transferred from one node to another, and its links can rotate between nodes. Through analyzing their symmetry properties, we can get the detail behavior of DCT, which can be simulated by computer with my program. And by comparing with space with 3 dimensions or 26 dimensions as string theory, we can get its CPT properties, which can be evidence of the theory.

**Category:** Topology

[36] **viXra:1402.0155 [pdf]**
*submitted on 2014-02-24 00:25:37*

**Authors:** S.Kalimuthu

**Comments:** 3 Pages. NA

In this work, a new concept for the creation of a new field of topology has been introduced.

**Category:** Topology

[35] **viXra:1401.0026 [pdf]**
*submitted on 2014-01-04 13:08:10*

**Authors:** Luis Sancho

**Comments:** 180 Pages.

The Universe is made of two formal motions, energy and information and its combinations, whose forms correspond to the 3 canonical topologies of a 4 dimensional Universe:
Hyperbolic, informative systems; energetic, spherical limbs, and toroid combinations of both.
The 3 systems together form complementary systems made of informative particle/heads, reproductive body/waves and energetic limbs/fields.
Each topology dominates an age or horizon of evolution of the species: the young, energetic age; the reproductive maturity and the informative 3rd age.
Further on individual systems gather in herds, cellular clusters and social networks, evolving into bigger systems, creating a social arrow common to all systems that evolve from simple particles into atoms, molecules, cells, organisms and super-organisms (biological systems) or matter states, celestial bodies, solar systems and galaxies.
Those simple principles of General Systems Sciences and Topology reduce the possibilities of morphological evolution explaining how complex forms such as informative eyes or energetic limbs can emerge, solving the old problem already considered by Darwin; and explain the fractal, self-similar organization of physical systems.

**Category:** Topology

[34] **viXra:1311.0115 [pdf]**
*submitted on 2013-11-16 15:08:15*

**Authors:** Nathan O. Schmidt

**Comments:** 17 pages, 3 figures, submitted to Algebras, Groups and Geometries

In this work, we deploy Santilli's iso-dual iso-topic lifting and Inopin's holographic ring (IHR) topology as a platform to introduce and assemble a tesseract from two inter-locking, iso-morphic, iso-dual cubes in Euclidean triplex space. For this, we prove that such an "iso-dual tesseract'' can be constructed by following a procedure of simple, flexible, topologically-preserving instructions. Moreover, these novel results are significant because the tesseract's state and structure are directly inferred from the one initial cube (rather than two distinct cubes), which identifies a new iso-geometrical inter-connection between Santilli's exterior and interior dynamical systems.

**Category:** Topology

[33] **viXra:1311.0059 [pdf]**
*submitted on 2013-11-09 05:09:59*

**Authors:** Dmitri Martila

**Comments:** 1 Page.

Simple but strict talk about homeomorphism in topology. The World. Shape. The Beginning and End are shown.

**Category:** Topology

[32] **viXra:1311.0031 [pdf]**
*submitted on 2013-11-04 15:20:13*

**Authors:** Nathan O. Schmidt

**Comments:** 12 pages, 2 figures, submitted to the Hadronic Journal

In this cutting-edge exploration, we introduce and define the "dynamic iso-sphere Inopin holographic ring" (IHR), which is built from a "dynamic iso-topic lifting" equipped with an iso-unit function that is characterized by constant change. The resulting developments indicate that the dynamic iso-sphere IHR is simultaneously iso-dual to an "exterior dynamic iso-sphere IHR" and an "interior dynamic iso-sphere IHR". For this, we identify both the continuously-varying and discretely-varying cases. Ultimately, the conclusions suggest that a new branch of iso-mathematics may be in order.

**Category:** Topology

[31] **viXra:1311.0030 [pdf]**
*submitted on 2013-11-04 15:45:13*

**Authors:** Reza Katebi, Nathan O. Schmidt

**Comments:** 11 pages, 1 figure, submitted to the Hadronic Journal

In this introductory paper, we use Santilli's iso-topic lifting as a platform to explore Mandelbrot's set. The objective is to upgrade Mandelbrot's complex quadratic polynomial with iso-multiplication and then probe the effects on this revolutionary fractal. For this, we define the "iso-complex quadratic polynomial" and engage it to generate an array of "Mandelbrot iso-sets" by varying the iso-unit. The computational results indicate two general topological effects: scale-deformation and boundary-deformation, which are consequently connected to dynamic iso-spaces. In total, these new and preliminary developments spark further insight into the emerging realm of iso-fractals.

**Category:** Topology

[30] **viXra:1311.0016 [pdf]**
*submitted on 2013-11-02 15:03:09*

**Authors:** Nathan O. Schmidt

**Comments:** 34 pages, 7 figures, accepted in the Hadronic Journal

We propose a preliminary framework that engages iso-triplex numbers and deformation order parameters to encode the spatial states of Iso Open Topological Strings (Iso-OTS) for fermions and the temporal states of Iso Closed Topological Strings (Iso-CTS) for bosons, where space and time are iso-dual. The objective is to introduce an elementary Topological Iso-String Theory (TIST) that complies with the holographic principle and fundamentally represents the twisting, winding, and deforming of helical, spiral, and vortical information structures---by default---for attacking superfluidic motion patterns and energy states with iso-topic lifting. In general, these preliminary results indicate a cutting-edge, flexible, consistent, and powerful iso-mathematical framework with considerable representational capability that warrants further examination, collaboration, construction, and discipline.

**Category:** Topology

[29] **viXra:1310.0219 [pdf]**
*submitted on 2013-10-24 17:06:41*

**Authors:** A. A. Salama, Mohamed Eisa, Florentin Smarandache

**Comments:** 2 Pages. In this paper we generalize the classical topological spaces to the notion of neutrosophic classical set. Finally, we construct the basic concepts of the neutrosophic classical topology and we obtain several properties. Possible application to GIS topolog

In this paper we generalize the classical topological spaces to the notion of neutrosophic classical set. Finally, we construct the basic concepts of the neutrosophic classical topology and we obtain several properties. Possible application to GIS topology rules are touched upon.

**Category:** Topology

[28] **viXra:1310.0218 [pdf]**
*submitted on 2013-10-24 17:08:56*

**Authors:** A. A. Salama, Mohamed Eisa, Florentin Smarandache

**Comments:** 2 Pages. In this paper we introduce the notion of filters on neutrosophic classical set which is considered as a generalization of filters studies, the important neutrosophic classical filters has been given. Several relations between different neutrosophic classi

In this paper we introduce the notion of filters on neutrosophic classical set which is considered as a generalization of filters studies, the important neutrosophic classical filters has been given. Several relations between different neutrosophic classical filters and neutrosophic topologies are also studied here. Possible applications to database systems are touched upon.

**Category:** Topology

[27] **viXra:1310.0216 [pdf]**
*submitted on 2013-10-24 17:16:23*

**Authors:** A. A. Salama, Mohamed Eisa, Florentin Smarandache

**Comments:** 2 Pages.

In this paper we introduce the notion of ideals on neutrosophic set which is considered as a generalization of fuzzy and fuzzy intuitionistic ideals studies in [9,11] , the important topological neutrosophic ideals has been given in [4]. The concept of neutrosophic local function is also introduced for a neutrosophic topological space. These concepts are discussed with a view to find new neutrosophic topology from the original one in [8]. The basic structure, especially a basis for such generated neutrosophic topologies and several relations between different topological neutrosophic ideals and neutrosophic topologies are also studied here. Possible application to GIS topology rules are touched upon.

**Category:** Topology

[26] **viXra:1310.0198 [pdf]**
*submitted on 2013-10-23 00:00:41*

**Authors:** Nathan O. Schmidt

**Comments:** 9 pages, accepted in the Hadronic Journal

In this exploration, we introduce and define "dynamic iso-spaces", which are cutting-edge iso-mathematical constructions that are built with "dynamic iso-topic liftings" for "dynamic iso-unit functions". For this, we consider both the continuous and discrete cases. Subsequently, we engineer two simple examples that engage Fibonacci's sequence and Mandelbrot's set to define a "Fibonacci dynamic iso-space" and a "Mandelbrot dynamic iso-space", respectively. In total, this array of resulting iso-structures indicates that a new branch of iso-mathematics may be in order.

**Category:** Topology

[25] **viXra:1310.0096 [pdf]**
*submitted on 2013-10-13 17:21:15*

**Authors:** A.A. Salama, S.A. Alblowi

**Comments:** 5 Pages. Neutrosophy has been introduced by Smarandache [7, 8] as a new branch of philosophy. The purpose of this paper is to construct a new set theory called the neutrosophic set. After given the fundamental definitions of neutrosophic set operations, we obtain

Neutrosophy has been introduced by Smarandache [7, 8] as a new branch of philosophy. The
purpose of this paper is to construct a new set theory called the neutrosophic set. After given the fundamental
definitions of neutrosophic set operations, we obtain several properties, and discussed the relationship between
neutrosophic sets and others. Finally, we extend the concepts of fuzzy topological space [4], and intuitionistic
fuzzy topological space [5, 6] to the case of neutrosophic sets. Possible application to superstrings and
space–time are touched upon.

**Category:** Topology

[24] **viXra:1310.0092 [pdf]**
*submitted on 2013-10-13 07:16:37*

**Authors:** A.A.Salama

**Comments:** 7 Pages. Fuzzy ideals and the notion of fuzzy local function were introduced and studied by Sarkar[12] and by Mahmoud in [9]. The purpose of this paper deals with a fuzzy compactness modulo a fuzzy ideal. Many new sorts of weak and strong fuzzy compactness have be

Fuzzy ideals and the notion of fuzzy local function were introduced and studied by Sarkar[12] and by Mahmoud in [9]. The purpose of this paper deals with a fuzzy compactness modulo a fuzzy ideal. Many new sorts of weak and strong fuzzy compactness have been introduced to fuzzy topological spaces in the last twenty years but not have been studied using fuzzy ideals so,the main aim of our work in this paper is to define and study some new various types of fuzzy compactness with respect to fuzzy ideals namely fuzzy L-compact and L*-compact spaces. Also fuzzy compactness with respect to ideal is useful as unification and generalization of several others widely studied concepts. Possible application to superstrings and E∞ space-time are touched upon.

**Category:** Topology

[23] **viXra:1310.0089 [pdf]**
*submitted on 2013-10-13 07:29:00*

**Authors:** M. E. Abd El-Monsef, A.Kozae, A. A. Salama, H.M.Elagmy

**Comments:** 5 Pages. In this paper we introduce the notion of fuzzy bitopological ideals .The concept of fuzzy pairwise local function is also introduced here by utilizing the q-neighborhood structure for a fuzzy topological space .These concepts are discussed fuzzy bitopolog

In this paper we introduce the notion of fuzzy bitopological ideals .The concept of fuzzy pairwise
local function is also introduced here by utilizing the q-neighborhood structure for a fuzzy topological space
.These concepts are discussed fuzzy bitopologies and several relations between different fuzzy bitopological
ideals .

**Category:** Topology

[22] **viXra:1310.0087 [pdf]**
*submitted on 2013-10-13 07:38:01*

**Authors:** A.A.Salama

**Comments:** 4 Pages. In this paper, fuzzy L-open sets due to Abd El-Monsef et al. [4] are used to introduce a new separation axiom and new type of function in fuzzy topological ideals spaces . Some the basic properties of fuzzy L-irresolute functions, as well as the connectio

In this paper, fuzzy L-open sets due to Abd El-Monsef et al. [4] are used to introduce a new
separation axiom and new type of function in fuzzy topological ideals spaces . Some the basic properties of fuzzy
L-irresolute functions, as well as the connections between them, are investigated. Possible application to
superstrings and space–time are touched upon.

**Category:** Topology

[21] **viXra:1310.0086 [pdf]**
*submitted on 2013-10-13 07:40:55*

**Authors:** M. E. Abd El-Monsef, A. Kozae, A. A. Salama, H. M. Elagamy

**Comments:** 4 Pages. The aim of this paper is to introduce and study some new fuzzy pairwise notion in fuzzy bitopological ideals spaces. We also generalize the notion of fuzzy L-open sets due to Abd El-Monsef et al[1]. In addit ion to generalize the concept of fuzzy L-closed

The aim of this paper is to introduce and study some new fuzzy pairwise notion in fuzzy bitopological ideals
spaces. We also generalize the notion of fuzzy L-open sets due to Abd El-Monsef et al[1]. In addit ion to generalize the
concept of fuzzy L-closed sets, fuzzy L-continuity and L-open functions due to Abd El-Monsef et al[1]. Relationships
between the above new fuzzy pairwise notions and there other relevant classes are investigated. Recently, we define and
study two different types of fuzzy pairwise functions .

**Category:** Topology

[20] **viXra:1310.0085 [pdf]**
*submitted on 2013-10-13 07:43:52*

**Authors:** A. A. Salama, S. A. Alblowi

**Comments:** 5 Pages. In this paper we introduce the notion of generalized intuitionistic fuzzy ideals which is considered as a generalization of fuzzy intuitionistic ideals studies in[6], the important generalized intuitionistic fuzzy ideals has been given. The concept of gen

In this paper we introduce the notion of generalized intuitionistic fuzzy ideals which is considered as a generalization of fuzzy intuitionistic ideals studies in[6], the important generalized intuitionistic fuzzy ideals has been given. The concept of generalized intuitionistic fuzzy local function is also introduced for a generalized intuitionistic fuzzy topological space. These concepts are discussed with a view to find new generalized intuitionistic fuzzy topology from the original one in[5, 7]. The basic structure, especially a basis for such generated generalized intuitionistic fuzzy topologies and several relations between different generalized intuitionistic fuzzy ideals and generalized intuitionistic fuzzy topologies are also studied here.

**Category:** Topology

[19] **viXra:1310.0084 [pdf]**
*submitted on 2013-10-13 07:48:25*

**Authors:** A.A. Salama, S.A. Alblowi

**Comments:** 10 Pages. In this paper we introduce the notion of intuitionistic fuzzy ideals which is considered as a generalization of fuzzy ideals studies in [1, 2, 3, 11 ], the important intuitionistic fuzzy ideal has been given. The concept of intuitionistic fuzzy local func

In this paper we introduce the notion of intuitionistic fuzzy ideals which is
considered as a generalization of fuzzy ideals studies in [1, 2, 3, 11 ], the
important intuitionistic fuzzy ideal has been given. The concept of
intuitionistic fuzzy local function is also introduced here by utilizing the -
neighborhood structure for an intuitionistic fuzzy topological space. These
concepts are discussed with a view to find new intuitionistic fuzzy topology
from the original one in [10, 12].The basic structure, especially a basis for
such generated intuitionistic fuzzy topologies and several relations between
different intuitionistic fuzzy ideals and intuitionistic fuzzy topologies are also
studied here. Finally, several properties of all investigated new notions are
discussed.

**Category:** Topology

[18] **viXra:1310.0083 [pdf]**
*submitted on 2013-10-13 07:50:40*

**Authors:** M.e. Abd El-Monsef, A.a. Nasef, A.a. Salama

**Comments:** 11 Pages. Recently in 1997, Sarker in [8] introduced the concept of fuzzy ideal and fuzzy local function between fuzzy topological spaces. In the present paper, we introduce some new fuzzy notions via fuzzy ideals. Also, we generalize the notion of L-open sets due

Recently in 1997, Sarker in [8] introduced the concept of
fuzzy ideal and fuzzy local function between fuzzy topological spaces.
In the present paper, we introduce some new fuzzy notions via fuzzy
ideals. Also, we generalize the notion of L-open sets due to Jankovic
and Homlett [6]. In addition to, we generalize the concept of L-closed
sets, L- continuity due to Abd El-Monsef et al. [2]. Relationships
between the above new fuzzy notions and other relevant classes are
investigated.1

**Category:** Topology

[17] **viXra:1310.0079 [pdf]**
*submitted on 2013-10-12 19:35:26*

**Authors:** A. A. Salama, S. A. Alblowi

**Comments:** 4 Pages.

In this paper we introduce definitions of generalized neutrosophic sets. After given the fundamental definitions
of generalized neutrosophic set operations, we obtain several properties, and discussed the relationship between generalized
neutrosophic sets and others. Finally, we extend the concepts of neutrosophic topological space [9], intuitionistic fuzzy
topological space [5, 6], and fuzzy topological space [4] to the case of generalized neutrosophic sets. Possible application to
GIS topology rules are touched upon.

**Category:** Topology

[16] **viXra:1308.0116 [pdf]**
*submitted on 2013-08-21 18:51:33*

**Authors:** John Frederick Sweeney

**Comments:** 35 Pages.

Polytope (3,3,5) plays an extremely crucial role in the transformation of visible matter, as well as in the structure of Time. Polytope (3,3,5) helps to determine whether matter follows the 8 x 8 Satva path or the 9 x 9 Raja path of development. Polytope (3,3,5) on a micro scale determines the development path of matter, while Polytope (3,3,5) on a macro scale determines the geography of Time, given its relationship to Base 60 math and to the icosahedron. Yet the Hopf Fibration is needed to form Poytope (3,3,5). This paper outlines the series of interchanges between root lattices and the three types of Hopf Fibrations in the formation of quasi – crystals.

**Category:** Topology

[15] **viXra:1308.0099 [pdf]**
*submitted on 2013-08-19 12:04:41*

**Authors:** John Frederick Sweeney

**Comments:** 64 Pages.

Mathematicians and physicists have long wondered why the Octionic Projective Plane (OP2), the Freudenthal – Tits Magic Square, or Magic Triangle and certain functions of the Octonions and Sedenions abruptly end. This paper lays out the various elements included in these conundra, with the assumption that irregularities and undiscovered relationships between these structures account for the anomalies. In addition to the above, this paper investigates the G2 to B3 to D4 to B4 to F4 Magic Triangle, the twisted product of S7 x S7 x G2, which leads to the Sedenions, the exceptional singularities, Kleinian singularities, Coxeter Groups H3 and H4, Polytope (3,3,5) , the 600 – cell and the binary icosahedral group.

**Category:** Topology

[14] **viXra:1308.0061 [pdf]**
*submitted on 2013-08-11 10:33:40*

**Authors:** John Frederick Sweeney

**Comments:** 82 Pages.

The final element of the Qi Men Dun Jia Model is the Boerdijk-Coxeter Helix, or the Tetrahelix of R. Buckminster Fuller, since this brings matter up to the level of DNA strings or lattices. Composed of Sextonions,Octonions, Twisted Octonions and Sedenions, the author examines the Boerdijk-Coxeter Helix from various perspectives to illustrate how BC – Helices play an important role in the formation of matter. The paper closely examines the eccentricities of the BC – Helix to determine whether these relate to diminished Octionic and Sedenion function, associativity and divisibility.

**Category:** Topology

[13] **viXra:1308.0026 [pdf]**
*submitted on 2013-08-05 15:28:53*

**Authors:** Alexander Fedorov

**Comments:** 25 Pages.

One of causes why Twin Primes problem was
unsolved over a long period is that
pairs of Twin Primes (PTP) are considered separately
from other pairs of Twin Numbers (PTN).
By purpose of this work is research of connections
between different types of PTN. For realization of this
purpose by author was developed the "Arithmetic of
pairs of Twin Numbers" (APTN).
In APTN are defined three types PTN.
As shown in APTN all types PTN are connected with
each other by relations which represent distribution of
prime and composite positive integers less than 2n
between them.
On the basis of this relations (axioms APTN) are
deduced formulas for computation of the number of PTN
(NPTN) for each types.
In APTN also is defined and computed Average value
of the number of pairs are formed from odd prime
and composite positive integers $ < 2n $ . Separately
AVNPP for prime and AVNPC for composite positive integers.
We also deduced formulas for computation of deviation
NPTN from AVNPP and AVNPC.
It was shown that if $n$ go to infinity then NPTN go to AVNPC or AVNPP
respectively that permit to apply formulas for AVNPP and AVNPC
to computation of NPTN.
At the end is produced the proof of the Twin Primes
problem with help of APTN.
It is shown that if n go to infinity then NPTP go to infinity.

**Category:** Topology

[12] **viXra:1308.0015 [pdf]**
*submitted on 2013-08-03 11:51:04*

**Authors:** John Frederick Sweeney

**Comments:** 31 Pages.

In Vedic Nuclear Physics, the number 28 plays a key role, and this will be discussed in a future paper. The 28 aspects must be extruded or dispersed along structures which contain factors of 12 or 6. The Poincare Dodecahedral Space contains these factors and relates to the 120-element binary icosahedral group, which double covers the simple 60-element icosahedral group. This, in turn, enjoys isometric relationships to the 60 Jia Zi and the 60 Na Yin of Chinese metaphysics, function to add Five Element and temporal qualities to matter as it becomes visible, in the Qi Men Dun Jia Model.

**Category:** Topology

[11] **viXra:1307.0036 [pdf]**
*submitted on 2013-07-07 06:11:23*

**Authors:** Vyacheslav Telnin

**Comments:** 8 Pages.

This paper begins from ordinary one-dimensional number line. Then starts the infinite process
of forming the sequences of big and little numbers. This process leads to the formation of two one-dimensional lines: positive and negative numbers. After that begins the detailed examination of each big and each little number. That leads to knowledge that some positive
numbers coincide with some negative numbers. And that some big numbers coincide with some little and with some medium numbers. And some little numbers coincide with some medium numbers. To illustrate this process there are 9 diagrams in the paper. In order to reflect these coincidences it is necessary to use 2 dimensions, then 3 dimensions, and so on … .So we see that simple number line has very complex topological structure.

**Category:** Topology

[10] **viXra:1307.0010 [pdf]**
*submitted on 2013-07-02 11:23:51*

**Authors:** John Frederick Sweeney

**Comments:** 34 Pages.

Qi Men Dun Jia is based on the Clifford Clock, as well as an icosahedron. The purpose of the icosahedron relates to the Pisano Period, which has a limit of 60, or the periodicity of Fibonacci Numbers and the Fibonacci Spiral, which are related to the Golden Ratio and the Platonic Solids. In addition, the icosahedron forms an isomorphic relationship to the 60 Jia Zi and 60 Na Yin of Chinese metaphysics, which provide the entry point for the Five Elements into the formation of matter. The icosahedron is composed of three Golden Rectangles and is edged in the Golden Ratio. The three Golden Rectangles are directly related to three Fano Planes, which are composed of Octonions. Taken together, the Pisano Period, Fibonacci Numbers and the Golden Section outline the path of growth of matter in the universe. Trigonometric, elliptic and Jacobi functions lend the model additional types of periodicity. By following this natural order, the Qi Men Dun Jia model is capable of making accurate predictions about natural and human phenomena, which can be replicated by other analysts.

**Category:** Topology

[9] **viXra:1306.0221 [pdf]**
*submitted on 2013-06-26 22:32:47*

**Authors:** John Frederick Sweeney

**Comments:** 21 Pages.

The ancient Chinese divination method called Qi Men Dun Jia is mathematically based on the Clifford Clock, which is a ring of Clifford Algebras related through the Bott Periodicity Theorem. On top of this lies the icosahedra or A5 with its sixty or one hundred twenty elements. A5 in the model functions to account for Time, as well as the Five Elements and the Na Jia, additional elements of Chinese metaphysics used in divination. This icosahedra is composed of three Golden Rectangles. The Golden Rectangles are related to three Fano Planes and the octonions, which are in turn related to the Golden Section. This provides the basis for the Lie Algebra and lattice of E8, again with its own Golden Section properties.

**Category:** Topology

[8] **viXra:1306.0192 [pdf]**
*submitted on 2013-06-22 12:52:13*

**Authors:** Liu Ran

**Comments:** 20 Pages.

Four-color theorem is an interesting phenomenon, but there is a rule hidden the phenomenon. The biggest adjacent relationship on a surface decides how many color enough. K-color theorem is a deducing from it.

**Category:** Topology

[7] **viXra:1306.0187 [pdf]**
*submitted on 2013-06-21 08:24:13*

**Authors:** Kamran Alam Khan

**Comments:** 4 Pages. Published in International Journal of Mathematical Archive (IJMA)

The notion of a bitopological space as a triple (X,I_1,I_2), where X is a set and I_1and I_2are topologies on X, was first formulated by J.C.Kelly [5]. In this paper our aim is to introduce and study the notion of an N-topological space (X,I_1,I_2,………I_N). We first generalize the notion of an ordinary metric to n variables. This metric will be called K-metric. Then the notion of a quasi-pseudo-K-metric will be introduced. We then follow the approach of Kelly to introduce and study the notion of an N-topological space. An example for such a space is produced using chain topology. And finally we define and study some of the possible separation properties for N-topological spaces.

**Category:** Topology

[6] **viXra:1302.0039 [pdf]**
*submitted on 2013-02-06 20:33:41*

**Authors:** Nasir Germain

**Comments:** 4 Pages.

my new spin on mathematics

**Category:** Topology

[5] **viXra:1302.0011 [pdf]**
*submitted on 2013-02-02 07:56:10*

**Authors:** Jaivir Baweja

**Comments:** 2 Pages.

Let $M$ be a smooth manifold. In this paper we review the definition of a germ
and show that since it is an equivalence relation, the concept is only locally defined.

**Category:** Topology

[4] **viXra:1205.0082 [pdf]**
*submitted on 2012-05-20 11:09:22*

**Authors:** Terry Allen, Daniel Branscombe, Jim Bury

**Comments:** 50 Pages.

The musical staff notates Pitch Value Vectors whereas tablature, using fret numbers on string lines, denotes Position Value Vectors, forming a commuting algebra of Hilbert Spaces. In 2001 I demonstrated that music is semi-algebraic (Allen and Goudessenue).
Pitch Value Space is undefined without a connection to pitch, and when connected to pitch by a barycenter, becomes defined and complete.
A defined musical system must have at least 2 functions, the chromatic f(x) and the harmonic function g(x) that form a composite function with at most 1 common center (Music Multicentricity Theorem). Thus tonality is defined by the line of tonal projection that marries pitch to position to make a musical tone.
Since musical systems must have a tone generator (instrument or device) the music topos must be the triple composite function f⋅g⋅h where f(x) is a + b + c = 0 and g(x) > 0 is a scale center and h(x) > 0 an instrument center. A music cipher as defined here as an affine projection that marries R:Z pitch to position to compose a note [tone point as an orthonormal pair (position value, pitch value)]. The harmonic message is embedded in a musical system by the cipher which defines tonality, so that (harmony, tonality) is another orthonormal pair.
A cipher can also make a new note from one already known in a system.
The only algebraic operation in a musical topos is vector additions to a single barycenter according to a difference function defined by the complete lattice of the musical system, and according to the Boolean Arithmetic Operator of the Music Cipher which forms the geometry of tone value spaces by its prime ideals. The cipher model is therefore simple and natural compared to current music topology requiring two centers and several algebraic operators.
Music is composed by the finite union of notes and open intervals defined by the composite functions of the fundamental, the key, and the intonation algorithm.
Tonality, the sum total of every function, relation, and element in a musical system, is the same as the algebraic-logic interface (numeric key) of the pitch-position intonation algorithm that is precisely the triangle of cipher vectors formed between one logic and at least two algebraic sub lattices. The cipher vector defined by a complete musical lattice is also the same as the arithmetic tone values closure operator that defines tonal geometry. Specifically, the cipher is precisely the projection between the logic sub lattice and at least two algebraic sub lattices in the musical system, where the sub lattices all share the fundamental as 1 common center.
Therefore the cipher is equivalent to a point, a line, a triangle, and a sphere, reflections resulting from line-point duality in geometry. Without a common center for the R: Z cipher the musical clock is undefined: Euler's donut is dead.
The new model is a clock: the fundamental is the hour hand, the instrument position is the minute hand, and scale position is the third hand. Tonality, like time on the clock, is a vector as a composite of three functions with 1 fundamental in common. Therefore, tonality has at least two functions but at most one center.

**Category:** Topology

[3] **viXra:1205.0081 [pdf]**
*submitted on 2012-05-20 16:05:13*

**Authors:** Tuomas Korppi

**Comments:** 39 Pages.

A homology theory based on both near-standard and non-near-standard
microsimplices is constructed. Its basic properties, including
Eilenberg-Steenrod axioms for homology and continuity with respect to resolutions
of spaces, are proved.

**Category:** Topology

[2] **viXra:1003.0267 [pdf]**
*submitted on 30 Mar 2010*

**Authors:** Victor Porton

**Comments:** 4 pages

Considered convergence and limit for funcoids (a generalization of proximity spaces).
I also have defined (generalized) limit for arbitrary (not necessarily continuous)
functions under certain conditions.
This article is a part of my Algebraic General Topology research.

**Category:** Topology

[1] **viXra:1003.0192 [pdf]**
*submitted on 16 Mar 2010*

**Authors:** Victor Porton

**Comments:** 32 pages

It is a part of my Algebraic General Topology research.
In this article I introduce the concepts of funcoids which generalize proximity spaces
and reloids which generalize uniform spaces. The concept of funcoid is generalized concept
of proximity space, the concept of reloid is cleared from superfluous details (generalized)
concept of uniform space. Also funcoids and reloids are generalizations of binary relations
whose domains and ranges are filters (instead of sets).
That funcoids and reloids are common generalizations of both (proximity, pretopology,
uniform) spaces and of (multivalued) functions, makes this theory smart for analyzing
properties (e.g. continuousness) of functions on spaces. Also funcoids and reloids can be
considered as a generalization of (oriented) graphs, this provides us with a common
generalization of analysis and discrete mathematics.

**Category:** Topology

[42] **viXra:1705.0210 [pdf]**
*replaced on 2017-05-20 07:34:40*

**Authors:** Saenko V.I.

**Comments:** 3 Pages. This is the Russian version, the English one is directed to the peer-reviewed journal

It is proved that the irreducible map according to Franklin consists of 5 regions and, as a consequence, 4 colors are sufficient for colouring any map on the sphere

**Category:** Topology

[41] **viXra:1609.0084 [pdf]**
*replaced on 2017-03-06 13:46:39*

**Authors:** Mirosław J. Kubiak

**Comments:** 7 Pages.

Until the early twentieth century, the three-dimensional space and one-dimensional time were considered separate beings. In 1909, German mathematician H. Minkowski connected together space and time into single idea, creating a new the four-dimensional spacetime.
In this paper we proposed the extension of this idea by the connection together the Minkowski four-dimensional spacetime and the mass density into the single idea, creating a new entity: the four-dimensional spacetime with the mass density.

**Category:** Topology

[40] **viXra:1608.0003 [pdf]**
*replaced on 2016-08-02 04:07:46*

**Authors:** Richard L

**Comments:** 26 Pages. Forgot authors name, also reformatted

We thank Newton for inspiring strict adherence to hypotheses non-fingo, and claim reasonable a posteriori surety in positing the need for an Ontological-Phase Topological Field Theory (OPTFT) as the final step in describing the remaining requirements for bulk UQC. Let’s surmise with little doubt that a radical new theory needs to be correlated with the looming 3rd regime of Unified Field Mechanics (UFM). If the author knows one thing for sure, it is that gravity is not quantized! The physics community is so invested in quantizing the gravitational force that it could still be years away from this inevitable conclusion. There is still a serious conundrum to be dealt with however; discovery of the complex Manifold of Uncertainty (MOU), the associated ‘semi-quantum limit’ and the fact of a duality between Newton’s and Einstein’s gravity, may allow some sort of wave-particle-like duality with a quantal-like virtual graviton in the semi-quantum limit. Why mention the gravitational field? Relativistic information processing (RIP) introduces gravitational effects in the ‘parallel transport’ aspects of topological switching in branes. There are A and B type topological string theories, and a related Topological M-Theory with mirror symmetry, that are somewhat interesting especially since they allow sufficient dimensionality with Calabi-Yau mirror symmetric dual 3-tori perceived as essential elements for developing a UFM. But a distinction between these theories and the ontology of an energyless topological switching of information (Shannon related) through topological charge in brane dynamics, perhaps defined in a manner making correspondence to a higher dimensional (HD) de-Broglie-Bohm super-quantum potential synonymous with a 'Force of coherence' of the unified field is of interest. Thus the term 'OPTFT’ has been chosen to address this issue as best as the Zeitgeist is able to conceive at the time of writing…

**Category:** Topology

[39] **viXra:1506.0139 [pdf]**
*replaced on 2015-07-11 05:48:01*

**Authors:** Luis Sancho

**Comments:** 32 Pages.

We expand the holographic principle to explain the nature of the universe and all its fractal 2-manifold organisms. The simplest explanation of the complex universe departs from the holographic principle: information, form is bidimensional. Because all what we perceive is the 'cover' of things, its membranes. Thus in a 2-manifold, in a bidimensional world there are only 2 topologies, which correspond to the main parts of any system, the lineal, energetic and spherical informative dimensions or motions (since energy is just space in motion and time information cyclical form in motion). And its 'wavelike combinations, exi. And we shall call energy, to th expansive motions, which form limbs, often with hyperbolic topology (bilateral). And we shall call information to the implosive vortices, which form particles and heads, often of spherical geometry. And the space between them the 3 membrane-topology of the universe, its body wave, the ExI part of the system. Thus all what exists are polar systems, with 'energy fields/limbs' and informative heads/particles, exchange energy and information creating a 3rd element, 'waves-bodies'.
We use the formalism to unify all disciplines of human knowledge, showing all its species display 5 isomorphisms, derived of the 3 topologic varieties:
- isomorphism of 3 finite, diffeomorphic dimensions of space
- isomorphism of 3 time-ages/states, past- young energy, liquid balanced present and solid, future, wrinkled age
- isomorphism of 3 space-time topologies, energy, iteration and information
- isomorphism of 3 hierarchies of organisation, the quantum/cellular, i-ndividual/human and social/cosmic scale
- isomorphism of 3 actions, energy feeding, reproductive iteration and informative gauging.
Whereas the 4 dimensional universe is the holographic intersection of 2-manifold membranes, and the Whole Universe a fractal sum of all of them

**Category:** Topology

[38] **viXra:1409.0124 [pdf]**
*replaced on 2015-08-17 09:28:44*

**Authors:** Vincenzo Nardozza

**Comments:** 4 Pages.

By means of a computer, all the possible homogeneous compact generalised triangulations made up of a small number of 3-simplexes (from 1 to 3) have been classified in homology classes. The analysis shows that, with a small number of simplexes, it is already possible to build quite a large number of separate topological spaces.

**Category:** Topology

[37] **viXra:1409.0124 [pdf]**
*replaced on 2014-09-26 05:16:45*

**Authors:** Vincenzo Nardozza

**Comments:** 10 Pages.

By means of a computer, all the possible homogeneous compact Generalised Triangulations made up of a small number of 3-simplexes (from 1 to 3) have been classified in homology classes. The analysis shows that, with a small number of simplexes, it is already possible to build quite a large number of separate topological spaces.

**Category:** Topology

[36] **viXra:1311.0192 [pdf]**
*replaced on 2014-05-20 17:46:37*

**Authors:** Nathan O. Schmidt

**Comments:** 19 pages, 5 figures, accepted in Algebras, Groups and Geometries

In this preliminary work, we focus on a particular iso-geometrical, iso-topological facet of iso-mathematics by suggesting a developing, generalized approach for encoding the states and transitions of spherically-symmetric structures that vary in size. In particular, we introduce the notion of "effective iso-radius" to facilitate a heightened characterization of dynamic iso-sphere Inopin holographic rings (IHR) as they undergo "iso-transitions" between "iso-states". In essence, we propose the existence of "effective dynamic iso-sphere IHRs". In turn, this emergence drives the construction of a new "effective iso-state" platform to encode the generalized dynamics of such iso-complex, non-linear systems in a relatively straightforward approach of spherical-based iso-topic liftings. The initial results of this analysis are significant because they lead to alternative modes of research and application, and thereby pose the question: do these effective dynamic iso-sphere IHRs have application in physics and chemistry? Our hypothesis is: yes. To answer this inquiry and assess this conjecture, this developing work should be subjected to further scrutiny, collaboration, improvement, and hard work via the scientific method in order to advance it as such.

**Category:** Topology

[35] **viXra:1311.0192 [pdf]**
*replaced on 2014-05-16 15:02:07*

**Authors:** Nathan O. Schmidt

**Comments:** 18 pages, 5 figures, submitted to Algebras, Groups and Geometries

In this preliminary work, we focus on a particular iso-geometrical, iso-topological facet of iso-mathematics by suggesting a developing, generalized approach for encoding the states and transitions of spherically-symmetric structures that vary in size. In particular, we introduce the notion of "effective iso-radius" to facilitate a heightened characterization of dynamic iso-sphere Inopin holographic rings (IHR) as they undergo "iso-transitions" between "iso-states". In essence, we propose the existence of "effective dynamic iso-sphere IHRs". In turn, this emergence drives the construction of a new "effective iso-state" platform to encode the generalized dynamics of such iso-complex, non-linear systems in a relatively straightforward approach of spherical-based iso-topic liftings. The initial results of this analysis are significant because they lead to alternative modes of research and application, and thereby pose the question: do these effective dynamic iso-sphere IHRs have application in physics and chemistry? Our hypothesis is: yes. To answer this inquiry and assess this conjecture, this developing work should be subjected to further scrutiny, collaboration, improvement, and hard work via the scientific method in order to advance it as such.

**Category:** Topology

[34] **viXra:1311.0031 [pdf]**
*replaced on 2014-02-08 15:45:08*

**Authors:** Nathan O. Schmidt

**Comments:** 13 pages, 2 figures, accepted by the Hadronic Journal

In this preliminary work, we use a dynamic iso-unit function to iso-topically lift the "static" Inopin holographic ring (IHR) of the unit sphere to an interconnected pair of "dynamic iso-sphere IHRs" (iso-DIHR), where the IHR is simultaneously iso-dual to both a magnified "exterior iso-DIHR" and de-magnified ``interior iso-DIHR". For both the continuously-varying and discretely-varying cases, we define the dynamic iso-amplitude-radius of one iso-DIHR as being equivalent to the dynamic iso-amplitude-curvature of its counterpart, and conversely. These initial results support the hypothesis that a new IHR-based mode of iso-geometry and iso-topology may be in order, which is significant because the interior and exterior zones delineated by the IHR are fundamentally "iso-dual inverses" and may be inferred from one another.

**Category:** Topology

[33] **viXra:1311.0030 [pdf]**
*replaced on 2013-11-20 19:31:09*

**Authors:** Reza Katebi, Nathan O. Schmidt

**Comments:** 14 pages, 2 figures, accepted in the Hadronic Journal

In this introductory work, we use Santilli's iso-topic lifting as a cutting-edge platform to explore Mandelbrot's set. The objective is to upgrade Mandelbrot's complex quadratic polynomial with iso-multiplication and then computationally probe the effects on this revolutionary fractal. For this, we define the "iso-complex quadratic polynomial" and engage it to generate a locally iso-morphic array of "Mandelbrot iso-sets" by varying the iso-unit, where the connectedness property is topologically preserved in each case. The iso-unit broadens and strengthens the chaotic analysis, and authorizes an enhanced classification and demystification such complex systems because it equips us with an additional degree of freedom: the new Mandelbrot iso-set array is an improvement over the traditional Mandelbrot set because it is significantly more general. In total, the experimental results exemplify dynamic iso-spaces and indicate two modes of topological effects: scale-deformation and boundary-deformation. Ultimately, these new and preliminary developments spark further insight into the emerging realm of iso-fractals.

**Category:** Topology

[32] **viXra:1308.0051 [pdf]**
*replaced on 2013-10-11 17:31:47*

**Authors:** Nathan O. Schmidt, Reza Katebi

**Comments:** 34 pages, 7 figures, accepted in the Hadronic Journal

In a preliminary assessment, we begin to apply Santilli's iso-mathematics to triplex numbers, Euclidean triplex space, triplex fractals, and Inopin's 2-sphere holographic ring (HR) topology. In doing so, we successfully identify and define iso-triplex numbers for iso-fractal geometry in a Euclidean iso-triplex space that is iso-metrically equipped with an iso-2-sphere HR topology. As a result, we state a series of lemmas that aim to characterize these emerging iso-mathematical structures. These initial outcomes indicate that it may be feasible to engage this encoding framework to systematically attack a broad range of problems in the disciplines of science and mathematics, but a thorough, rigorous, and collaborative investigation should be in order to challenge, refine, upgrade, and implement these ideas.

**Category:** Topology

[31] **viXra:1307.0036 [pdf]**
*replaced on 2015-08-17 09:10:45*

**Authors:** Vyacheslav Telnin

**Comments:** 8 Pages.

This paper deals with the number line. Usually it is considered as one-dimensional object. But if
to take into account the infinite large and the infinite little numbers, then this line turns out to
be more complex object with infinite self-crossings in some many-dimensional space.

**Category:** Topology

[30] **viXra:1307.0036 [pdf]**
*replaced on 2013-09-07 05:33:42*

**Authors:** Vyacheslav Telnin

**Comments:** 8 Pages.

This paper deals with the number line. Usually it is considered as one-dimensional object. But if
to take into account the infinite large and the infinite little numbers, then this line turns out to be more complex object with infinite self-crossings in some many-dimensional space.

**Category:** Topology

[29] **viXra:1306.0192 [pdf]**
*replaced on 2013-09-03 08:53:52*

**Authors:** Liu Ran

**Comments:** 38 Pages.

The method and basic theory are far from traditional graph theory. Maybe they are the key factor of success. K4 regions (every region is adjacent to other 3 regions) are the max adjacent relationship, four-color theorem is true because more than 4 regions, there must be a non-adjacent region existing. Non-adjacent regions can be color by the same color and decrease color consumption.
Another important three-color theorem is that the border of regions can be colored by 3 colors. Every region has at least 2 optional colors, which can be permuted.

**Category:** Topology

[28] **viXra:1306.0192 [pdf]**
*replaced on 2013-07-29 10:00:10*

**Authors:** Liu Ran

**Comments:** 42 Pages.

The method and basic theory are far from traditional graph theory. Maybe they are the key factor of success. K4 regions (every region is adjacent to other 3 regions) are the max adjacent relationship, four-color theorem is true because more than 4 regions, there must be a non-adjacent region existing. Non-adjacent regions can be color by the same color and decrease color consumption.
Another important three-color theorem is that the border of regions can be colored by 3 colors. Every region has at least 2 optional colors, which can be permuted.

**Category:** Topology

[27] **viXra:1306.0192 [pdf]**
*replaced on 2013-07-27 21:41:36*

**Authors:** Liu Ran

**Comments:** 41 Pages.

The method and basic theory are far from traditional graph theory. Maybe they are the key factor of success. K4 regions (every region is adjacent to other 3 regions) are the max adjacent relationship, four-color theorem is true because more than 4 regions, there must be a non-adjacent region existing. Non-adjacent regions can be color by the same color and decrease color consumption.
Another important three-color theorem is that the border of regions can be colored by 3 colors. Every region has at least 2 optional colors, which can be permuted.

**Category:** Topology

[26] **viXra:1306.0192 [pdf]**
*replaced on 2013-07-27 10:49:06*

**Authors:** Liu Ran

**Comments:** 41 Pages.

**Category:** Topology

[25] **viXra:1306.0192 [pdf]**
*replaced on 2013-07-25 10:39:27*

**Authors:** Liu Ran

**Comments:** 35 Pages.

**Category:** Topology

[24] **viXra:1306.0192 [pdf]**
*replaced on 2013-07-18 10:30:04*

**Authors:** Liu Ran

**Comments:** 36 Pages.

Four-color theorem is an interesting phenomenon, but there is a rule hidden the phenomenon. The biggest adjacent relationship on a surface decides how many color enough. K-color theorem is a deducing from it.

**Category:** Topology

[23] **viXra:1306.0192 [pdf]**
*replaced on 2013-07-16 09:50:57*

**Authors:** Liu Ran

**Comments:** 33 Pages.

Four-color theorem is an interesting phenomenon, but there is a rule hidden the phenomenon. The biggest adjacent relationship on a surface decides how many color enough. K-color theorem is a deducing from it.

**Category:** Topology

[22] **viXra:1306.0192 [pdf]**
*replaced on 2013-07-16 08:35:28*

**Authors:** Liu Ran

**Comments:** 33 Pages.

**Category:** Topology

[21] **viXra:1306.0192 [pdf]**
*replaced on 2013-07-13 11:04:12*

**Authors:** Liu Ran

**Comments:** 32 Pages.

**Category:** Topology

[20] **viXra:1306.0192 [pdf]**
*replaced on 2013-07-03 10:48:17*

**Authors:** Liu Ran

**Comments:** 23 Pages.

**Category:** Topology

[19] **viXra:1306.0192 [pdf]**
*replaced on 2013-06-24 10:24:04*

**Authors:** Liu Ran

**Comments:** 23 Pages.

**Category:** Topology

[18] **viXra:1306.0192 [pdf]**
*replaced on 2013-06-23 10:58:19*

**Authors:** Liu Ran

**Comments:** 20 Pages.

**Category:** Topology

[17] **viXra:1003.0192 [pdf]**
*replaced on 19 Aug 2011*

**Authors:** Victor Porton

**Comments:** 53 pages

It is a part of my Algebraic General Topology research.
In this article, I introduce the concepts of funcoids, which generalize proximity spaces
and reloids, which generalize uniform spaces. The concept of funcoid is generalized concept
of proximity, the concept of reloid is cleared from superfluous details (generalized) concept of
uniformity. Also funcoids generalize pretopologies and preclosures. Also funcoids and reloids
are generalizations of binary relations whose domains and ranges are filters (instead of sets).
Also funcoids and reloids can be considered as a generalization of (oriented) graphs, this
provides us with a common generalization of analysis and discrete mathematics. The concept of continuity is defined by an algebraic formula (instead of old messy epsilondelta notation) for arbitrarymorphisms (including funcoids and reloids) of a partially ordered category. In one formula are generalized continuity, proximity continuity, and uniform continuity.

**Category:** Topology

[16] **viXra:1003.0192 [pdf]**
*replaced on 10 Aug 2011*

**Authors:** Victor Porton

**Comments:** 52 pages

It is a part of my Algebraic General Topology research.
In this article I introduce the concepts of funcoids which generalize proximity spaces
and reloids which generalize uniform spaces. The concept of funcoid is generalized concept
of proximity, the concept of reloid is cleared from superfluous details (generalized) concept
of uniformity. Also funcoids and reloids are generalizations of binary relations whose
domains and ranges are filters (instead of sets).
Also funcoids and reloids can be considered as a generalization of (oriented) graphs,
this provides us with a common generalization of analysis and discrete mathematics.
The concept of continuity is defined by an algebraic formula (instead of old messy
epsilon-delta notation) for arbitrary morphisms (including funcoids and reloids) of a partially
ordered category. In one formula are generalized continuity, proximity continuity,
and uniform continuity.

**Category:** Topology

[15] **viXra:1003.0192 [pdf]**
*replaced on 2 Aug 2011*

**Authors:** Victor Porton

**Comments:** 52 pages

It is a part of my Algebraic General Topology research.
In this article I introduce the concepts of funcoids which generalize proximity spaces
and reloids which generalize uniform spaces. The concept of funcoid is generalized concept
of proximity, the concept of reloid is cleared from superfluous details (generalized) concept
of uniformity. Also funcoids and reloids are generalizations of binary relations whose
domains and ranges are filters (instead of sets).
Also funcoids and reloids can be considered as a generalization of (oriented) graphs,
this provides us with a common generalization of analysis and discrete mathematics.
The concept of continuity is defined by an algebraic formula (instead of old messy
epsilon-delta notation) for arbitrary morphisms (including funcoids and reloids) of a partially
ordered category. In one formula are generalized continuity, proximity continuity,
and uniform continuity.

**Category:** Topology

[14] **viXra:1003.0192 [pdf]**
*replaced on 29 Jul 2011*

**Authors:** Victor Porton

**Comments:** 52 pages

It is a part of my Algebraic General Topology research.
In this article I introduce the concepts of funcoids which generalize proximity spaces
and reloids which generalize uniform spaces. The concept of funcoid is generalized concept
of proximity, the concept of reloid is cleared from superfluous details (generalized) concept
of uniformity. Also funcoids and reloids are generalizations of binary relations whose
domains and ranges are filters (instead of sets).
Also funcoids and reloids can be considered as a generalization of (oriented) graphs,
this provides us with a common generalization of analysis and discrete mathematics.
The concept of continuity is defined by an algebraic formula (instead of old messy
epsilon-delta notation) for arbitrary morphisms (including funcoids and reloids) of a partially
ordered category. In one formula are generalized continuity, proximity continuity,
and uniform continuity.

**Category:** Topology

[13] **viXra:1003.0192 [pdf]**
*replaced on 3 Dec 2010*

**Authors:** Victor Porton

**Comments:** 46 pages

**Category:** Topology

[12] **viXra:1003.0192 [pdf]**
*replaced on 2 Dec 2010*

**Authors:** Victor Porton

**Comments:** 45 pages

**Category:** Topology

[11] **viXra:1003.0192 [pdf]**
*replaced on 4 Nov 2010*

**Authors:** Victor Porton

**Comments:** 44 pages

**Category:** Topology

[10] **viXra:1003.0192 [pdf]**
*replaced on 2 Nov 2010*

**Authors:** Victor Porton

**Comments:** 44 pages

**Category:** Topology

[9] **viXra:1003.0192 [pdf]**
*replaced on 30 Oct 2010*

**Authors:** Victor Porton

**Comments:** 43 pages

**Category:** Topology

[8] **viXra:1003.0192 [pdf]**
*replaced on 28 Oct 2010*

**Authors:** Victor Porton

**Comments:** 42 pages

**Category:** Topology

[7] **viXra:1003.0192 [pdf]**
*replaced on 25 Sep 2010*

**Authors:** Victor Porton

**Comments:** 42 pages

**Category:** Topology

[6] **viXra:1003.0192 [pdf]**
*replaced on 21 Sep 2010*

**Authors:** Victor Porton

**Comments:** 41 pages

**Category:** Topology

[5] **viXra:1003.0192 [pdf]**
*replaced on 13 Jun 2010*

**Authors:** Victor Porton

**Comments:** 39 pages

**Category:** Topology

[4] **viXra:1003.0192 [pdf]**
*replaced on 21 Apr 2010*

**Authors:** Victor Porton

**Comments:** 39 pages

**Category:** Topology

[3] **viXra:1003.0192 [pdf]**
*replaced on 29 Mar 2010*

**Authors:** Victor Porton

**Comments:** 38 pages

**Category:** Topology

[2] **viXra:1003.0192 [pdf]**
*replaced on 26 Mar 2010*

**Authors:** Victor Porton

**Comments:** 37 pages

**Category:** Topology

[1] **viXra:1003.0192 [pdf]**
*replaced on 17 Mar 2010*

**Authors:** Victor Porton

**Comments:** 33 pages

It is a part of my Algebraic General Topology research.
In this article I introduce the concepts of funcoids which generalize proximity spaces
and reloids which generalize uniform spaces. The concept of funcoid is generalized concept
of proximity space, the concept of reloid is cleared from superfluous details (generalized)
concept of uniform space. Also funcoids and reloids are generalizations of binary relations
whose domains and ranges are filters (instead of sets).
Also funcoids and reloids can be considered as a generalization of (oriented) graphs,
this provides us with a common generalization of analysis and discrete mathematics.
The concept of continuity is defined by an algebraic formula (instead of old messy
epsilon-delta notation) for arbitrary morphisms (including funcoids and reloids) of a
partially ordered category. In one formula are generalized continuity, proximity continuity,
and uniform continuity.

**Category:** Topology