Статті в журналах: "Finite topological spaces"

1

Benoumhani, Moussa, and Ali Jaballah. "Finite fuzzy topological spaces." Fuzzy Sets and Systems 321 (August 2017): 101–14. http://dx.doi.org/10.1016/j.fss.2016.11.003.

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2

OSAKI, Takao. "Reduction of Finite Topological Spaces." Interdisciplinary Information Sciences 5, no. 2 (1999): 149–55. http://dx.doi.org/10.4036/iis.1999.149.

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3

Chae, Hi-joon. "FINITE TOPOLOGICAL SPACES AND GRAPHS." Communications of the Korean Mathematical Society 32, no. 1 (January 31, 2017): 183–91. http://dx.doi.org/10.4134/ckms.c160004.

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4

Bagchi, Susmit. "Topological Sigma-Semiring Separation and Ordered Measures in Noetherian Hyperconvexes." Symmetry 14, no. 2 (February 20, 2022): 422. http://dx.doi.org/10.3390/sym14020422.

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The interplay between topological hyperconvex spaces and sigma-finite measures in such spaces gives rise to a set of analytical observations. This paper introduces the Noetherian class of k-finite k-hyperconvex topological subspaces (NHCs) admitting countable finite covers. A sigma-finite measure is constructed in a sigma-semiring in a NHC under a topological ordering of NHCs. The topological ordering relation maintains the irreflexive and anti-symmetric algebraic properties while retaining the homeomorphism of NHCs. The monotonic measure sequence in a NHC determines the convexity and compactness of topological subspaces. Interestingly, the topological ordering in NHCs in two isomorphic topological spaces induces the corresponding ordering of measures in sigma-semirings. Moreover, the uniform topological measure spaces of NHCs need not always preserve the pushforward measures, and a NHC semiring is functionally separable by a set of inner-measurable functions.

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5

Edelsbrunner, Herbert, and Nimish R. Shah. "Triangulating Topological Spaces." International Journal of Computational Geometry & Applications 07, no. 04 (August 1997): 365–78. http://dx.doi.org/10.1142/s0218195997000223.

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Given a subspace [Formula: see text] and a finite set S⊆ℝd, we introduce the Delaunay complex, [Formula: see text], restricted by [Formula: see text]. Its simplices are spanned by subsets T⊆S for which the common intersection of Voronoi cells meets [Formula: see text] in a non-empty set. By the nerve theorem, [Formula: see text] and [Formula: see text] are homotopy equivalent if all such sets are contractible. This paper proves a sufficient condition for [Formula: see text] and [Formula: see text] be homeomorphic.

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6

Clader, Emily. "Inverse limits of finite topological spaces." Homology, Homotopy and Applications 11, no. 2 (2009): 223–27. http://dx.doi.org/10.4310/hha.2009.v11.n2.a11.

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7

Nakasho, Kazuhisa, Hiroyuki Okazaki, and Yasunari Shidama. "Finite Dimensional Real Normed Spaces are Proper Metric Spaces." Formalized Mathematics 29, no. 4 (December 1, 2021): 175–84. http://dx.doi.org/10.2478/forma-2021-0017.

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Summary In this article, we formalize in Mizar [1], [2] the topological properties of finite-dimensional real normed spaces. In the first section, we formalize the Bolzano-Weierstrass theorem, which states that a bounded sequence of points in an n-dimensional Euclidean space has a certain subsequence that converges to a point. As a corollary, it is also shown the equivalence between a subset of an n-dimensional Euclidean space being compact and being closed and bounded. In the next section, we formalize the definitions of L1-norm (Manhattan Norm) and maximum norm and show their topological equivalence in n-dimensional Euclidean spaces and finite-dimensional real linear spaces. In the last section, we formalize the linear isometries and their topological properties. Namely, it is shown that a linear isometry between real normed spaces preserves properties such as continuity, the convergence of a sequence, openness, closeness, and compactness of subsets. Finally, it is shown that finite-dimensional real normed spaces are proper metric spaces. We referred to [5], [9], and [7] in the formalization.

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8

K. K, Bushra Beevi, and Baby Chacko. "PARACOMPACTNESS IN GENERALIZED TOPOLOGICAL SPACES." South East Asian J. of Mathematics and Mathematical Sciences 19, no. 01 (April 30, 2023): 287–300. http://dx.doi.org/10.56827/seajmms.2023.1901.24.

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In this paper we introduce the concepts G - locally finite, σG - locally finite and G - paracompactness. Also discuss about some properties of these concepts. Here we investigate that some properties in topological spaces and generalized topological spaces (GTS) are coincides if we replace open sets by generalized open sets (G - open sets ). Also, we provide some examples to show some results are invalid in the case of GTS.

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9

Kang, Jeong, Sang-Eon Han, and Sik Lee. "The Fixed Point Property of Non-Retractable Topological Spaces." Mathematics 7, no. 10 (September 21, 2019): 879. http://dx.doi.org/10.3390/math7100879.

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Unlike the study of the fixed point property (FPP, for brevity) of retractable topological spaces, the research of the FPP of non-retractable topological spaces remains. The present paper deals with the issue. Based on order-theoretic foundations and fixed point theory for Khalimsky (K-, for short) topological spaces, the present paper studies the product property of the FPP for K-topological spaces. Furthermore, the paper investigates the FPP of various types of connected K-topological spaces such as non-K-retractable spaces and some points deleted K-topological (finite) planes, and so on. To be specific, after proving that not every one point deleted subspace of a finite K-topological plane X is a K-retract of X, we study the FPP of a non-retractable topological space Y, such as one point deleted space Y ∖ { p } .

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10

Nogin, Maria, and Bing Xu. "Modal Logic Axioms Valid in Quotient Spaces of Finite CW-Complexes and Other Families of Topological Spaces." International Journal of Mathematics and Mathematical Sciences 2016 (2016): 1–3. http://dx.doi.org/10.1155/2016/9163014.

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In this paper we consider the topological interpretations of L□, the classical logic extended by a “box” operator □ interpreted as interior. We present extensions of S4 that are sound over some families of topological spaces, including particular point topological spaces, excluded point topological spaces, and quotient spaces of finite CW-complexes.

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11

Bagchi, Susmit. "Projective and Non-Projective Varieties of Topological Decomposition of Groups with Embeddings." Symmetry 12, no. 3 (March 12, 2020): 450. http://dx.doi.org/10.3390/sym12030450.

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In general, the group decompositions are formulated by employing automorphisms and semidirect products to determine continuity and compactification properties. This paper proposes a set of constructions of novel topological decompositions of groups and analyzes the behaviour of group actions under the topological decompositions. The proposed topological decompositions arise in two varieties, such as decomposition based on topological fibers without projections and decomposition in the presence of translated projections in topological spaces. The first variety of decomposition introduces the concepts of topological fibers, locality of group operation and the partitioned local homeomorphism resulting in formulation of transitions and symmetric surjection within the topologically decomposed groups. The reformation of kernel under decomposed homeomorphism and the stability of group action with the existence of a fixed point are analyzed. The first variety of decomposition does not require commutativity maintaining generality. The second variety of projective topological decomposition is formulated considering commutative as well as noncommutative projections in spaces. The effects of finite translations of topologically decomposed groups under projections are analyzed. Moreover, the embedding of a decomposed group in normal topological spaces is formulated in this paper. It is shown that Schoenflies homeomorphic embeddings preserve group homeomorphism in the decomposed embeddings within normal topological spaces. This paper illustrates that decomposed group embedding in normal topological spaces is separable. The applications aspects as well as parametric comparison of group decompositions based on topology, direct product and semidirect product are included in the paper.

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12

Larose, Benoit, and L�szl� Z�dori. "Finite posets and topological spaces in locally finite varieties." algebra universalis 52, no. 2-3 (January 2005): 119–36. http://dx.doi.org/10.1007/s00012-004-1819-7.

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13

Bagchi, Susmit. "On the Analysis and Computation of Topological Fuzzy Measure in Distributed Monoid Spaces." Symmetry 11, no. 1 (December 22, 2018): 9. http://dx.doi.org/10.3390/sym11010009.

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The computational applications of fuzzy sets are pervasive in systems with inherent uncertainties and multivalued logic-based approximations. The existing fuzzy analytic measures are based on regularity variations and the construction of fuzzy topological spaces. This paper proposes an analysis of the general fuzzy measures in n-dimensional topological spaces with monoid embeddings. The embedded monoids are topologically distributed in the measure space. The analytic properties of compactness and homeomorphic, as well as isomorphic maps between spaces, are presented. The computational evaluations are carried out with n = 1, considering a set of translation functions with different symmetry profiles. The results illustrate the dynamics of finite fuzzy measure in a monoid topological subspace.

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14

Ochiai, Shoji. "On a topological invariant of finite topological spaces and enumerations." Tsukuba Journal of Mathematics 16, no. 1 (June 1992): 63–74. http://dx.doi.org/10.21099/tkbjm/1496161830.

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15

Orsatti, A., and N. Rodinò. "Homeomorphisms between finite powers of topological spaces." Topology and its Applications 23, no. 3 (August 1986): 271–77. http://dx.doi.org/10.1016/0166-8641(85)90044-6.

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16

Helmstutler, Randall D., and Ryan S. Higginbottom. "Finite Topological Spaces as a Pedagogical Tool." PRIMUS 22, no. 1 (January 2012): 64–74. http://dx.doi.org/10.1080/10511970.2010.492493.

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17

Ma, Liwen. "Important matrix computations in finite topological spaces." Applied Mathematics and Computation 395 (April 2021): 125808. http://dx.doi.org/10.1016/j.amc.2020.125808.

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18

Bagchi, Susmit. "Interactions between Homotopy and Topological Groups in Covering (C, R) Space Embeddings." Symmetry 13, no. 8 (August 3, 2021): 1421. http://dx.doi.org/10.3390/sym13081421.

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The interactions between topological covering spaces, homotopy and group structures in a fibered space exhibit an array of interesting properties. This paper proposes the formulation of finite covering space components of compact Lindelof variety in topological (C, R) spaces. The covering spaces form a Noetherian structure under topological injective embeddings. The locally path-connected components of covering spaces establish a set of finite topological groups, maintaining group homomorphism. The homeomorphic topological embedding of covering spaces and base space into a fibered non-compact topological (C, R) space generates two classes of fibers based on the location of identity elements of homomorphic groups. A compact general fiber gives rise to the discrete variety of fundamental groups in the embedded covering subspace. The path-homotopy equivalence is admitted by multiple identity fibers if, and only if, the group homomorphism is preserved in homeomorphic topological embeddings. A single identity fiber maintains the path-homotopy equivalence in the discrete fundamental group. If the fiber is an identity-rigid variety, then the fiber-restricted finite and symmetric translations within the embedded covering space successfully admits path-homotopy equivalence involving kernel. The topological projections on a component and formation of 2-simplex in fibered compact covering space embeddings generate a prime order cyclic group. Interestingly, the finite translations of the 2-simplexes in a dense covering subspace assist in determining the simple connectedness of the covering space components, and preserves cyclic group structure.

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19

Smyth, M. B., and J. Webster. "Finite approximation of stably compact spaces." Applied General Topology 3, no. 2 (October 1, 2002): 197. http://dx.doi.org/10.4995/agt.2002.2063.

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Finite approximation of spaces by inverse sequences of graphs (in the category of so-called topological graphs) was introduced by Smyth, and developed further. The idea was subsequently taken up by Kopperman and Wilson, who developed their own purely topological approach using inverse spectra of finite T0-spaces in the category of stably compact spaces. Both approaches are, however, restricted to the approximation of (compact) Hausdorff spaces and therefore cannot accommodate, for example, the upper space and (multi-) function space constructions. We present a new method of finite approximation of stably compact spaces using finite stably compact graphs, which when the topology is discrete are simply finite directed graphs. As an extended example, illustrating the problems involved, we study (ordered spaces and) arcs.

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20

Muradov, F. Kh. "Ternary semigroups of topological transformations." BULLETIN OF THE KARAGANDA UNIVERSITY-MATHEMATICS 102, no. 2 (June 30, 2021): 84–91. http://dx.doi.org/10.31489/2021m2/84-91.

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A ternary semigroup is a nonempty set with a ternary operation which is associative. The purpose of the present paper is to give a characterization of open sets of finite-dimensional Euclidean spaces by ternary semigroups of pairs of homeomorphic transformations and extend to ternary semigroups certain results of L.M. Gluskin concerned with semigroups of homeomorphic transformations of finite-dimensional Euclidean spaces.

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21

Boonpok, Chawalit. "Semi- I -Expandable Ideal Topological Spaces." Journal of Mathematics 2021 (December 6, 2021): 1–7. http://dx.doi.org/10.1155/2021/9272335.

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Our purpose is to introduce the notion of semi- I -expandable ideal topological spaces. Some properties of semi- I -locally finite collections are investigated. In particular, several characterizations of semi- I -expandable ideal topological spaces are established.

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22

Li, Chang-Qing, and Yan-Lan Zhang. "On characterizations of finite topological spaces with granulation and evidence theory." Filomat 33, no. 19 (2019): 6425–33. http://dx.doi.org/10.2298/fil1919425l.

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The theory of finite topological spaces can be used to investigate deep well-known problems in Topology, Algebra, Geometry and Artificial Intelligence. To represent uncertainty knowledge of a finite topological space, two kinds of measurement of a finite topological space are first introduced. Firstly, a kind of granularity of a finite topological space is defined, and properties of the granularity are explored. Secondly, relationships between the belief and plausibility functions in the Dempser-Shafer theory of evidence and the interior and closure operators in topological theory are established. The probabilities of interior and closure of sets construct a pair of belief and plausibility functions and its belief structure. And, for a belief structure with some properties, there exists a probability and a finite topology such that the belief and plausibility functions defined by the given belief structure are, respectively, the belief and plausibility functions by the topology. Then a necessary and sufficient condition for a belief structure to be the belief structure induced by a finite topology is presented.

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23

Leiderman, Arkady, and Sidney Morris. "Separability of Topological Groups: A Survey with Open Problems." Axioms 8, no. 1 (December 29, 2018): 3. http://dx.doi.org/10.3390/axioms8010003.

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Separability is one of the basic topological properties. Most classical topological groups and Banach spaces are separable; as examples we mention compact metric groups, matrix groups, connected (finite-dimensional) Lie groups; and the Banach spaces C ( K ) for metrizable compact spaces K; and ℓ p , for p ≥ 1 . This survey focuses on the wealth of results that have appeared in recent years about separable topological groups. In this paper, the property of separability of topological groups is examined in the context of taking subgroups, finite or infinite products, and quotient homomorphisms. The open problem of Banach and Mazur, known as the Separable Quotient Problem for Banach spaces, asks whether every Banach space has a quotient space which is a separable Banach space. This paper records substantial results on the analogous problem for topological groups. Twenty open problems are included in the survey.

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24

Shirazi, Zadeh, and Nasser Golestani. "On classifications of transformation semigroups: Indicator sequences and indicator topological spaces." Filomat 26, no. 2 (2012): 313–29. http://dx.doi.org/10.2298/fil1202313s.

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In this paper considering a transformation semigroup with finite height we define the notion of indicator sequence in such a way that any two transformation semigroups with the same indicator sequence have the same height. Also related to any transformation semigroup a topological space, called indicator topological space, is defined in such a way that transformation semigroups with homeomorphic indicator topological spaces have the same height. Moreover any two transformation semigroups with homeomorphic indicator topological spaces and finite height have the same indicator sequences.

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25

Kiliçman, Adem, and Anwar Jabor Fawakhreh. "Product Property on Generalized Lindelöf Spaces." ISRN Mathematical Analysis 2011 (April 6, 2011): 1–7. http://dx.doi.org/10.5402/2011/843480.

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We study the product properties of nearly Lindelöf, almost Lindelöf, and weakly Lindelöf spaces. We prove that in weak P-spaces, these topological properties are preserved under finite topological products. We also show that the product of separable spaces is weakly Lindelöf.

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26

Blanc, Anthony. "Topological K-theory of complex noncommutative spaces." Compositio Mathematica 152, no. 3 (September 22, 2015): 489–555. http://dx.doi.org/10.1112/s0010437x15007617.

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The purpose of this work is to give a definition of a topological K-theory for dg-categories over$\mathbb{C}$and to prove that the Chern character map from algebraic K-theory to periodic cyclic homology descends naturally to this new invariant. This topological Chern map provides a natural candidate for the existence of a rational structure on the periodic cyclic homology of a smooth proper dg-algebra, within the theory of noncommutative Hodge structures. The definition of topological K-theory consists in two steps: taking the topological realization of algebraic K-theory and inverting the Bott element. The topological realization is the left Kan extension of the functor ‘space of complex points’ to all simplicial presheaves over complex algebraic varieties. Our first main result states that the topological K-theory of the unit dg-category is the spectrum$\mathbf{BU}$. For this we are led to prove a homotopical generalization of Deligne’s cohomological proper descent, using Lurie’s proper descent. The fact that the Chern character descends to topological K-theory is established by using Kassel’s Künneth formula for periodic cyclic homology and the proper descent. In the case of a dg-category of perfect complexes on a separated scheme of finite type, we show that we recover the usual topological K-theory of complex points. We show as well that the Chern map tensorized with$\mathbb{C}$is an equivalence in the case of a finite-dimensional associative algebra – providing a formula for the periodic homology groups in terms of the stack of finite-dimensional modules.

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27

Mukharjee, A. "A new approach to nearly paracompact spaces." Matematychni Studii 60, no. 2 (December 18, 2023): 201–7. http://dx.doi.org/10.30970/ms.60.2.201-207.

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The pre-open sets are a generalization of open sets of topological spaces. In this paper, we introduce and study a notion of po-paracompact spaces via pre-open sets on topological spaces. We see that po-paracompact spaces are equivalent to nearly paracompact spaces. However, we find new characterizations to nearly paracompact spaces when we study it in the sense of poparacompact spaces. We see that a topological space is nearly paracompact if and only if each regularly open cover of the topological space has a locally finite pre-open refinement. We also show that four statements involving pre-open sets on an almost regular topological space are equivalent. A result on a subspace of a topological space is also obtained in term of pre-open sets.

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28

Shravan, Karishma, and Binod Chandra Tripathy. "Metrizability of multiset topological spaces." SERIES III - MATEMATICS, INFORMATICS, PHYSICS 13(62), no. 2 (January 20, 2021): 683–96. http://dx.doi.org/10.31926/but.mif.2020.13.62.2.24.

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In this paper, we have investigated one of the basic topological properties, called Metrizability in multiset topological space. Metrizable spaces are those topological spaces which are homeomorphic to a metric space. So, we first give the notion of metric between two multi-points in a finite multiset and studied some significant properties of a multiset metric space. The notion of metrizability is then studied by using this metric. Besides, the Urysohn’s lemma which is considered to be one of the important tools in studying some metrization theorems in topology is also discussed in context with multisets.

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29

Han, Sang-Eon. "Properties of space set topological spaces." Filomat 30, no. 9 (2016): 2475–87. http://dx.doi.org/10.2298/fil1609475h.

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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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30

Al-shami, T. M. "Compactness on Soft Topological Ordered Spaces and Its Application on the Information System." Journal of Mathematics 2021 (January 17, 2021): 1–12. http://dx.doi.org/10.1155/2021/6699092.

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It is well known every soft topological space induced from soft information system is soft compact. In this study, we integrate between soft compactness and partially ordered set to introduce new types of soft compactness on the finite spaces and investigate their application on the information system. First, we initiate a notion of monotonic soft sets and establish its main properties. Second, we introduce the concepts of monotonic soft compact and ordered soft compact spaces and show the relationships between them with the help of examples. We give a complete description for each one of them by making use of the finite intersection property. Also, we study some properties associated with some soft ordered spaces and finite product spaces. Furthermore, we investigate the conditions under which these concepts are preserved between the soft topological ordered space and its parametric topological ordered spaces. In the end, we provide an algorithm for expecting the missing values of objects on the information system depending on the concept of ordered soft compact spaces.

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31

KELLERER, HANS G., and G. WINKLER. "RANDOM DYNAMICAL SYSTEMS ON ORDERED TOPOLOGICAL SPACES." Stochastics and Dynamics 06, no. 03 (September 2006): 255–300. http://dx.doi.org/10.1142/s0219493706001797.

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Let (Xn, n ≥ 0) be a random dynamical system and its state space be endowed with a reasonable topology. Instead of completing the structure as common by some linearity, this study stresses — motivated in particular by economic applications — order aspects. If the underlying random transformations are supposed to be order-preserving, this results in a fairly complete theory. First of all, the classical notions of and familiar criteria for recurrence and transience can be extended from discrete Markov chain theory. The most important fact is provided by the existence and uniqueness of a locally finite-invariant measure for recurrent systems. It allows to derive ergodic theorems as well as to introduce an attract or in a natural way. The classification is completed by distinguishing positive and null recurrence corresponding, respectively, to the case of a finite or infinite invariant measure; equivalently, this amounts to finite or infinite mean passage times. For positive recurrent systems, moreover, strengthened versions of weak convergence as well as generalized laws of large numbers are available.

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32

Ayadi, Mohamed. "Twisted pre-Lie algebras of finite topological spaces." Communications in Algebra 50, no. 5 (November 22, 2021): 2115–38. http://dx.doi.org/10.1080/00927872.2021.1999461.

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33

Chen, Piwei, Hongliang Lai, and Dexue Zhang. "Coreflective hull of finite strong L-topological spaces." Fuzzy Sets and Systems 182, no. 1 (November 2011): 79–92. http://dx.doi.org/10.1016/j.fss.2010.05.001.

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34

Clader, Emily. "Erratum to “Inverse limits of finite topological spaces”." Homology, Homotopy and Applications 18, no. 1 (2016): 25–26. http://dx.doi.org/10.4310/hha.2016.v18.n1.a2.

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35

Eguchi, Masayoshi, Hiroshi Imura, Yasushi Fuwa, and Yatsuka Nakamura. "Digital straight line segments in finite topological spaces." Electronics and Communications in Japan (Part III: Fundamental Electronic Science) 81, no. 2 (February 1998): 34–45. http://dx.doi.org/10.1002/(sici)1520-6440(199802)81:2<34::aid-ecjc4>3.0.co;2-5.

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36

Baek, Cheol Woo, Jang Hyun Jo, and Yeong Seong Jo. "Classification of complete regularities for finite topological spaces." Acta Mathematica Hungarica 137, no. 3 (September 5, 2012): 153–57. http://dx.doi.org/10.1007/s10474-012-0255-y.

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37

Alkouri, Abd Ulazeez, Mohammad Hazaimeh, and Ibrahim Jawarneh. "More on Fuzzy Topological Spaces on Fuzzy Space." International Journal of Fuzzy Systems and Advanced Applications 8 (August 5, 2021): 21–26. http://dx.doi.org/10.46300/91017.2021.8.2.

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The fuzzy topological space was introduced by Dip in 1999 depending on the notion of fuzzy spaces. Dip’s approach helps to rectify the deviation in some definitions of fuzzy subsets in fuzzy topological spaces. In this paper, further definitions, and theorems on fuzzy topological space fill the lack in Dip’s article. Different types of fuzzy topological space on fuzzy space are presented such as co-finite, co-countable, right and left ray, and usual fuzzy topology. Furthermore, boundary, exterior, and isolated points of fuzzy sets are investigated and illustrated based on fuzzy spaces. Finally, separation axioms are studied on fuzzy spaces

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38

Künzi, Hans-Peter A. "Some remarks on quasi-uniform spaces." Glasgow Mathematical Journal 31, no. 3 (September 1989): 309–20. http://dx.doi.org/10.1017/s0017089500007874.

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A topological space is called a uqu space [10] if it admits a unique quasi-uniformity. Answering a question [2, Problem B, p. 45] of P. Fletcher and W. F. Lindgren in the affirmative we show in [8] that a topological space X is a uqu space if and only if every interior-preserving open collection of X is finite. (Recall that a collection ℒ of open sets of a topological space is called interior-preserving if the intersection of an arbitrary subcollection of ℒ is open (see e.g. [2, p. 29]).) The main step in the proof of this result in [8] shows that a topological space in which each interior-preserving open collection is finite is a transitive space. (A topological space is called transitive (see e.g. [2, p. 130]) if its fine quasi-uniformity has a base consisting of transitive entourages.) In the first section of this note we prove that each hereditarily compact space is transitive. The result of [8] mentioned above is an immediate consequence of this fact, because, obviously, a topological space in which each interior-preserving open collection is finite is hereditarily compact; see e.g. [2, Theorem 2.36]. Our method of proof also shows that a space is transitive if its fine quasi-uniformity is quasi-pseudo-metrizable. We use this result to prove that the fine quasi-uniformity of a T1 space X is quasi-metrizable if and only if X is a quasi-metrizable space containing only finitely many nonisolated points. This result should be compared with Proposition 2.34 of [2], which says that the fine quasi-uniformity of a regular T1 space has a countable base if and only if it is a metrizable space with only finitely many nonisolated points (see e.g. [11] for related results on uniformities). Another by-product of our investigations is the result that each topological space with a countable network is transitive.

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39

Ercolessi, Elisa, Giovanni Landi, and Paulo Teotonio-Sobrinho. "Noncommutative Lattices and the Algebras of Their Continuous Functions." Reviews in Mathematical Physics 10, no. 04 (May 1998): 439–66. http://dx.doi.org/10.1142/s0129055x98000148.

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Recently a new kind of approximation to continuum topological spaces has been introduced, the approximating spaces being partially ordered sets (posets) with a finite or at most a countable number of points. The partial order endows a poset with a nontrivial non-Hausdorff topology. Their ability to reproduce important topological information of the continuum has been the main motivation for their use in quantum physics. Posets are truly noncommutative spaces, or noncommutative lattices, since they can be realized as structure spaces of noncommutative C*-algebras. These noncommutative algebras play the same rôle as the algebra of continuous functions [Formula: see text] on a Hausdorff topological space M and can be thought of as algebras of operator valued functions on posets. In this article, we will review some mathematical results that establish a duality between finite posets and a certain class of C*-algebras. We will see that the algebras in question are all postliminal approximately finite dimensional (AF) algebras.

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40

Bors, Cristina, María V. Ferrer, and Salvador Hernández. "Bounded Sets in Topological Spaces." Axioms 11, no. 2 (February 10, 2022): 71. http://dx.doi.org/10.3390/axioms11020071.

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Let G be a monoid that acts on a topological space X by homeomorphisms such that there is a point x0∈X with GU=X for each neighbourhood U of x0. A subset A of X is said to be G-bounded if for each neighbourhood U of x0 there is a finite subset F of G with A⊆FU. We prove that for a metrizable and separable G-space X, the bounded subsets of X are completely determined by the bounded subsets of any dense subspace. We also obtain sufficient conditions for a G-space X to be locally G-bounded, which apply to topological groups. Thereby, we extend some previous results accomplished for locally convex spaces and topological groups.

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41

Khalimsky, Efim. "Topological structures in computer science." Journal of Applied Mathematics and Simulation 1, no. 1 (January 1, 1987): 25–40. http://dx.doi.org/10.1155/s1048953388000036.

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Topologies of finite spaces and spaces with countably many points are investigated. It is proven, using the theory of ordered topological spaces, that any topology in connected ordered spaces, with finitely many points or in spaces similar to the set of all integers, is an interval-alternating topology. Integer and digital lines, arcs, and curves are considered. Topology of N-dimensional digital spaces is described. A digital analog of the intermediate value theorem is proven. The equivalence of connectedness and pathconnectedness in digital and integer spaces is also proven. It is shown here how methods of continuous mathematics, for example, topological methods, can be applied to objects, that used to be investigated only by methods of discrete mathematics. The significance of methods and ideas in digital image and picture processing, robotic vision, computer tomography and system's sciences presented here is well known.

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42

Gabriyelyan, Saak S., and Sidney A. Morris. "Free Subspaces of Free Locally Convex Spaces." Journal of Function Spaces 2018 (2018): 1–5. http://dx.doi.org/10.1155/2018/2924863.

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IfXandYare Tychonoff spaces, letL(X)andL(Y)be the free locally convex space overXandY, respectively. For generalXandY, the question of whetherL(X)can be embedded as a topological vector subspace ofL(Y)is difficult. The best results in the literature are that ifL(X)can be embedded as a topological vector subspace ofL(I), whereI=[0,1], thenXis a countable-dimensional compact metrizable space. Further, ifXis a finite-dimensional compact metrizable space, thenL(X)can be embedded as a topological vector subspace ofL(I). In this paper, it is proved thatL(X)can be embedded inL(R)as a topological vector subspace ifXis a disjoint union of a countable number of finite-dimensional locally compact separable metrizable spaces. This is the case ifX=Rn, n∈N.It is also shown that ifGandQdenote the Cantor space and the Hilbert cubeIN, respectively, then (i)L(X)is embedded inL(G)if and only ifXis a zero-dimensional metrizable compact space; (ii)L(X)is embedded inL(Q)if and only ifYis a metrizable compact space.

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43

Richmond, Thomas A. "Finite-point order compactifications." Mathematical Proceedings of the Cambridge Philosophical Society 102, no. 3 (November 1987): 467–73. http://dx.doi.org/10.1017/s0305004100067529.

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After the characterization of 1-point topological compactifications by Alexandroff in 1924, n-point topological compactifications by Magill [4] in 1965, and 1-point order compactifications by McCallion [5] in 1971, spaces that admit an n -point order compactification are characterized in Section 2. If X* and X** are finite-point order compactifications of X, sup{X*, X**} is given explicitly in terms of X* and X** in § 3. In § 4 it is shown that if an ordered topological space X has an m-point and an n-point order compactification, then X has a k-point order compactification for each integer k between m and n. The author is indebted to Professor Darrell C. Kent, who provided assistance and encouragement during the preparation of this paper.

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44

LEIDERMAN, ARKADY, and SIDNEY A. MORRIS. "EMBEDDINGS OF FREE TOPOLOGICAL VECTOR SPACES." Bulletin of the Australian Mathematical Society 101, no. 2 (August 20, 2019): 311–24. http://dx.doi.org/10.1017/s000497271900090x.

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It is proved that the free topological vector space $\mathbb{V}([0,1])$ contains an isomorphic copy of the free topological vector space $\mathbb{V}([0,1]^{n})$ for every finite-dimensional cube $[0,1]^{n}$, thereby answering an open question in the literature. We show that this result cannot be extended from the closed unit interval $[0,1]$ to general metrisable spaces. Indeed, we prove that the free topological vector space $\mathbb{V}(X)$ does not even have a vector subspace isomorphic as a topological vector space to $\mathbb{V}(X\oplus X)$, where $X$ is a Cook continuum, which is a one-dimensional compact metric space. This is also shown to be the case for a rigid Bernstein set, which is a zero-dimensional subspace of the real line.

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45

Han, Sang-Eon. "Hereditary properties of semi-separation axioms and their applications." Filomat 32, no. 13 (2018): 4689–700. http://dx.doi.org/10.2298/fil1813689h.

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The paper studies the open-hereditary property of semi-separation axioms and applies it to the study of digital topological spaces such as an n-dimensional Khalimsky topological space, a Marcus-Wyse topological space and so on. More precisely, we study various properties of digital topological spaces related to low-level and semi-separation axioms such as T1/2 , semi-T1/2 , semi-T1, semi-T2, etc. Besides, using the finite or the infinite product property of the semi-Ti-separation axiom, i ? {1,2}, we prove that the n-dimensional Khalimsky topological space is a semi-T2-space. After showing that not every subspace of the digital topological spaces satisfies the semi-Ti-separation axiom, i ?{1,2}, we prove that the semi-Tiseparation property is open-hereditary, i ? {1,2}. All spaces in the paper are assumed to be nonempty and connected.

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46

Sari, Dewi Kartika, and Dongsheng Zhao. "A new cardinal function on topological spaces." Applied General Topology 18, no. 1 (April 3, 2017): 75. http://dx.doi.org/10.4995/agt.2017.5869.

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Using neighbourhood assignments, we introduce and study a new cardinal function, namely GCI(X), for every topological space X. We shall mainly investigate the spaces X with finite GCI(X). Some properties of this cardinal in connection with special types of mappings are also proved.

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47

Künzi, Hans-Peter A. "Topological Spaces with a Unique Compatible Quasi-Uniformity." Canadian Mathematical Bulletin 29, no. 1 (March 1, 1986): 40–43. http://dx.doi.org/10.4153/cmb-1986-007-3.

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48

Mejías, Luis, Jorge Vielma, Ángel Guale, and Ebner Pineda. "Primal Topologies on Finite-Dimensional Vector Spaces Induced by Matrices." International Journal of Mathematics and Mathematical Sciences 2023 (January 6, 2023): 1–7. http://dx.doi.org/10.1155/2023/9393234.

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Given an matrix A , considered as a linear map A : ℝ n ⟶ ℝ n , then A induces a topological space structure on ℝ n which differs quite a lot from the usual one (induced by the Euclidean metric). This new topological structure on ℝ n has very interesting properties with a nice special geometric flavor, and it is a particular case of the so called “primal space,” In particular, some algebraic information can be shown in a topological fashion and the other way around. If X is a non-empty set and f : X ⟶ X is a map, there exists a topology τ f induced on X by f , defined by τ f = U ⊂ X : f − 1 U ⊂ U . The pair X , τ f is called the primal space induced by f . In this paper, we investigate some characteristics of primal space structure induced on the vector space ℝ n by matrices; in particular, we describe geometrical properties of the respective spaces for the case.

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49

Tanaka, Kohei. "A combinatorial description of topological complexity for finite spaces." Algebraic & Geometric Topology 18, no. 2 (March 12, 2018): 779–96. http://dx.doi.org/10.2140/agt.2018.18.779.

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50

Charatonik, J. J., and W. J. Charatonik. "Generalized homogeneity of finite and of countable topological spaces." Rocky Mountain Journal of Mathematics 18, no. 1 (March 1988): 195–210. http://dx.doi.org/10.1216/rmj-1988-18-1-195.

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