Remarks on topological spaces on $ {\mathbb Z}^n $ which are related to the Khalimsky $ n $-dimensional space

In the orientiworld framework the Standard Model fields are localized on D3-branes sitting on top of an orientifold three-plane. The transverse six-dimensional space is a noncompact orbifold (or a more general conifold). The four-dimensional gravity on D3-branes is reproduced due to the four-dimensional Einstein–Hilbert term induced at the quantum level. The orientifold three-plane plays a crucial role, in particular, without it the D3-brane world-volume theories would be conformal due to the tadpole cancellation. We study nonperturbative gauge dynamics in various [Formula: see text] supersymmetric orientiworld models based on the Z3 as well as Z5 and Z7 orbifold groups. Our discussions illustrate that there is a rich variety of supersymmetry preserving dynamics in some of these models. On the other hand, we also find some models with dynamical supersymmetry breaking.

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Complexity for partial computable functions over computable Polish spaces

Mathematical Structures in Computer Science ◽

10.1017/s0960129516000438 ◽

2016 ◽

Vol 28 (3) ◽

pp. 429-447 ◽

Cited By ~ 3

Author(s):

MARGARITA KOROVINA ◽

OLEG KUDINOV

Keyword(s):

Crucial Role ◽

Polish Space ◽

Computable Function ◽

Topological Spaces ◽

Computable Numbering ◽

Polish Spaces ◽

The Real ◽

Computable Functions

In the framework of effectively enumerable topological spaces, we introduce the notion of a partial computable function. We show that the class of partial computable functions is closed under composition, and the real-valued partial computable functions defined on a computable Polish space have a principal computable numbering. With respect to the principal computable numbering of the real-valued partial computable functions, we investigate complexity of important problems such as totality and root verification. It turns out that for some problems the corresponding complexity does not depend on the choice of a computable Polish space, whereas for other ones the corresponding choice plays a crucial role.

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Free Abelian topological groups and the Pontryagin-Van Kampen duality

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700014726 ◽

1995 ◽

Vol 52 (2) ◽

pp. 297-311 ◽

Cited By ~ 20

Author(s):

Vladimir Pestov

Keyword(s):

Topological Group ◽

Dimensional Space ◽

Topological Spaces ◽

Topological Groups ◽

Abelian Topological Group ◽

Free Abelian Topological Group

We study the class of Tychonoff topological spaces such that the free Abelian topological group A(X) is reflexive (satisfies the Pontryagin-van Kampen duality). Every such X must be totally path-disconnected and (if it is pseudocompact) must have a trivial first cohomotopy group π1(X). If X is a strongly zero-dimensional space which is either metrisable or compact, then A(X) is reflexive.

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Topological graphs based on a new topology on Zn and its applications

Filomat ◽

10.2298/fil1720313h ◽

2017 ◽

Vol 31 (20) ◽

pp. 6313-6328 ◽

Cited By ~ 5

Author(s):

Sang-Eon Han

Keyword(s):

Extension Problem ◽

Topological Spaces ◽

Natural Numbers ◽

Topological Graph ◽

Topological Graphs ◽

Graph Homomorphism ◽

Continuous Map ◽

Higher Dimensional ◽

Khalimsky Topology ◽

Digital Spaces

Up to now there is no homotopy for Marcus-Wyse (for short M-) topological spaces. In relation to the development of a homotopy for the category of Marcus-Wyse (for short M-) topological spaces on Z2, we need to generalize the M-topology on Z2 to higher dimensional spaces X ? Zn, n ? 3 [18]. Hence the present paper establishes a new topology on Zn; n 2 N, where N is the set of natural numbers. It is called the generalized Marcus-Wyse (for short H-) topology and is denoted by (Zn, n). Besides, we prove that (Z3, 3) induces only 6- or 18-adjacency relations. Namely, (Z3, 3) does not support a 26-adjacency, which is quite different from the Khalimsky topology for 3D digital spaces. After developing an H-adjacency induced by the connectedness of (Zn; n), the present paper establishes topological graphs based on the H-topology, which is called an HA-space, so that we can establish a category of HA-spaces. By using the H-adjacency, we propose an H-topological graph homomorphism (for short HA-map) and an HA-isomorphism. Besides, we prove that an HA-map (resp. an HA-isomorphism) is broader than an H-continuous map (resp. an Hhomeomorphism) and is an H-connectedness preserving map. Finally, after investigating some properties of an HA-isomorphism, we propose both an HA-retract and an extension problem of an HA-map for studying HA-spaces.

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The Sequential and Contractible Topological Embeddings of Functional Groups

Symmetry ◽

10.3390/sym12050789 ◽

2020 ◽

Vol 12 (5) ◽

pp. 789

Author(s):

Susmit Bagchi

Keyword(s):

Topological Space ◽

Functional Groups ◽

Dimensional Space ◽

Rotational Symmetry ◽

Topological Spaces ◽

Algebraic Construction ◽

Closed Curves ◽

Existing Structures ◽

The One ◽

One Point Compactification

The continuous and injective embeddings of closed curves in Hausdorff topological spaces maintain isometry in subspaces generating components. An embedding of a circle group within a topological space creates isometric subspace with rotational symmetry. This paper introduces the generalized algebraic construction of functional groups and its topological embeddings into normal spaces maintaining homeomorphism of functional groups. The proposed algebraic construction of functional groups maintains homeomorphism to rotationally symmetric circle groups. The embeddings of functional groups are constructed in a sequence in the normal topological spaces. First, the topological decomposition and associated embeddings of a generalized group algebraic structure in the lower dimensional space is presented. It is shown that the one-point compactification property of topological space containing the decomposed group embeddings can be identified. Second, the sequential topological embeddings of functional groups are formulated. The proposed sequential embeddings follow Schoenflies property within the normal topological space. The preservation of homeomorphism between disjoint functional group embeddings under Banach-type contraction is analyzed taking into consideration that the underlying topological space is Hausdorff and the embeddings are in a monotone class. It is shown that components in a monotone class of isometry are not separable, whereas the multiple disjoint monotone class of embeddings are separable. A comparative analysis of the proposed concepts and formulations with respect to the existing structures is included in the paper.

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An extension problem of a connectedness preserving map between Khalimsky spaces

Filomat ◽

10.2298/fil1601015h ◽

2016 ◽

Vol 30 (1) ◽

pp. 15-28 ◽

Cited By ~ 1

Author(s):

Sang-Eon Han

Keyword(s):

Computer Science ◽

Mathematical Morphology ◽

Digital Images ◽

Digital Topology ◽

Extension Problem ◽

Topological Spaces ◽

Digital Geometry ◽

Applied Topology ◽

Continuous Map ◽

Connected Sets

The goal of the present paper is to study an extension problem of a connected preserving (for short, CP-) map between Khalimsky (K-for brevity, if there is no ambiguity) spaces. As a generalization of a K-continuous map, for K-topological spaces the recent paper [13] develops a function sending connected sets to connected ones (for brevity, an A-map: see Definition 3.1 in the present paper). Since this map plays an important role in applied topology including digital topology, digital geometry and mathematical morphology, the present paper studies an extension problem of a CP-map in terms of both an A-retract and an A-isomorphism (see Example 5.2). Since K-topological spaces have been often used for studying digital images, this extension problem can contribute to a certain areas of computer science and mathematical morphology.

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On Homeomorphisms between Extension Spaces

Canadian Journal of Mathematics ◽

10.4153/cjm-1960-020-8 ◽

1960 ◽

Vol 12 ◽

pp. 252-262 ◽

Cited By ~ 2

Author(s):

Bernhard Banaschewski

Keyword(s):

General Result ◽

Hausdorff Space ◽

Dimensional Space ◽

Regular Space ◽

Topological Spaces ◽

Minimal Extension ◽

Completely Regular ◽

General Terms ◽

Extension Spaces ◽

Homeomorphic Extension

In this note, conditions are obtained which will ensure that two topological spaces are homeomorphic when they have homeomorphic extension spaces of a certain kind. To discuss this topic in suitably general terms, an unspecified extension procedure, assumed to be applicable to some class of topological spaces, is considered first, and it is shown that simple conditions imposed on the extension procedure and its domain of operation easily lead to a condition of the desired kind. After the general result has been established it is shown to be applicable to a number of particular extensions, such as the Stone-Čech compactification and the Hewitt Q-extension of a completely regular Hausdorff space, Katětov's maximal Hausdorff-closed extension of a Hausdorff space, the maximal zero-dimensional compactification of a zero-dimensional space, the maximal Hausdorff-minimal extension of a semi-regular space, and Freudenthal's compactification of a rim-compact space. The case of the Hewitt Q-extension was first discussed by Heider (6).

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Solve Special Case of Some Guran Problems

Journal of Mathematics Research ◽

10.5539/jmr.v8n3p33 ◽

2016 ◽

Vol 8 (3) ◽

pp. 33

Author(s):

Ahmad Alghoussein ◽

Ziad Kanaya ◽

Salwa Yacoub

Keyword(s):

Dimensional Space ◽

Topological Spaces ◽

Topological Groups ◽

Open Problems ◽

Hereditary Properties ◽

Special Case

Throughout this paper, all topological groups are assumed to be topological differential manifolds and algebraically free, our aim in this paper is to prove the open problems number (7) and (8). Which are introduced by (Guran, I, 1998). In many cases of spaces and under a suitable conditions. therefore, we denote by I(X) and I(Y) to be a free topological groups over a topological spaces X and Y respectively where X and Y are assumed to be a non- empty sub manifolds Which are also a closed sub sets, and P is a classes of topological spaces, as a regular, normal, Tychonoff, lindelöf, separable connected, compact and Zero- dimensional space, and we have tried to use a hereditary properties and others of these spaces, so we can prove the open problems in these cases and we have many results showed in this paper.

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Solve Special Case of Some Guran Problems

Journal of Mathematics Research ◽

10.5539/jmr.v8n4p33 ◽

2016 ◽

Vol 8 (3) ◽

pp. 33

Author(s):

Ahmad Alghoussein ◽

Ziad Kanaya ◽

Salwa Yacoub

Keyword(s):

Dimensional Space ◽

Topological Spaces ◽

Topological Groups ◽

Open Problems ◽

Hereditary Properties ◽

Special Case

Throughout this paper, all topological groups are assumed to be topological differential manifolds and algebraically free, our aim in this paper is to prove the open problems number (7) and (8). Which are introduced by (Guran, I, 1998). In many cases of spaces and under a suitable conditions. therefore, we denote by I(X) and I(Y) to be a free topological groups over a topological spaces X and Y respectively where X and Y are assumed to be a non- empty sub manifolds Which are also a closed sub sets, and P is a classes of topological spaces, as a regular, normal, Tychonoff, lindelöf, separable connected, compact and Zero- dimensional space, and we have tried to use a hereditary properties and others of these spaces, so we can prove the open problems in these cases and we have many results showed in this paper.

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Properties of space set topological spaces

Filomat ◽

10.2298/fil1609475h ◽

2016 ◽

Vol 30 (9) ◽

pp. 2475-2487 ◽

Cited By ~ 2

Author(s):

Sang-Eon Han

Keyword(s):

Topological Space ◽

Crucial Role ◽

Topological Structure ◽

Special Kind ◽

Topological Spaces ◽

Separation Axiom ◽

Locally Finite ◽

Alexandroff Space ◽

Finite Spaces ◽

Point Set

Since a locally finite topological structure plays an important role in the fields of pure and applied topology, the paper studies a special kind of locally finite spaces, so called a space set topology (for brevity, SST) and further, proves that an SST is an Alexandroff space satisfying the separation axiom T0. Unlike a point set topology, since each element of an SST is a space, the present paper names the topology by the space set topology. Besides, for a connected topological space (X,T) with |X| = 2 the axioms T0, semi-T1/2 and T1/2 are proved to be equivalent to each other. Furthermore, the paper shows that an SST can be used for studying both continuous and digital spaces so that it plays a crucial role in both classical and digital topology, combinatorial, discrete and computational geometry. In addition, a connected SST can be a good example showing that the separation axiom semi-T1/2 does not imply T1/2.

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