scholarly journals Study of a Forwarding Chain in the Category of Topological Spaces between T0 and T2 with respect to One Point Compactification Operator

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Fatemah Ayatollah Zadeh Shirazi ◽  
Meysam Miralaei ◽  
Fariba Zeinal Zadeh Farhadi
Keyword(s):  
Topological Space ◽  
Topological Spaces ◽  
The One ◽  
One Point Compactification

In the following text, we want to study the behavior of one point compactification operator in the chain Ξ := {Metrizable, Normal, T2, KC, SC, US, T1, TD, TUD, T0, Top} of subcategories of category of topological spaces, Top (where we denote the subcategory of Top, containing all topological spaces with property P , simply by P). Actually we want to know, for P∈Ξ and X∈P, the one point compactification of topological space X belongs to which elements of Ξ. Finally we find out that the chain {Metrizable, T2, KC, SC, US, T1, TD, TUD, T0, Top} is a forwarding chain with respect to one point compactification operator.

scholarly journals Embedding inverse semigroups of homeomorphisms on closed subsets

1977 ◽  
Vol 18 (2) ◽  
pp. 199-207 ◽  
Author(s):  
Bridget Bos Baird
Keyword(s):  
Topological Space ◽  
Inverse Semigroup ◽  
Inverse Semigroups ◽  
Topological Spaces ◽  
Locally Compact ◽  
Compact Metric Space ◽  
Countable Space ◽  
The One ◽  
One Point Compactification ◽  
First Countable Space

All topological spaces here are assumed to be T2. The collection F(Y)of all homeomorphisms whose domains and ranges are closed subsets of a topological space Y is an inverse semigroup under the operation of composition. We are interested in the general problem of getting some information about the subsemigroups of F(Y) whenever Y is a compact metric space. Here, we specifically look at the problem of determining those spaces X with the property that F(X) is isomorphic to a subsemigroup of F(Y). The main result states that if X is any first countable space with an uncountable number of points, then the semigroup F(X) can be embedded into the semigroup F(Y) if and only if either X is compact and Y contains a copy of X, or X is noncompact and locally compact and Y contains a copy of the one-point compactification of X.

scholarly journals The Sequential and Contractible Topological Embeddings of Functional Groups

Symmetry ◽  
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.

Essentials of General Topology

2021 ◽  
pp. 191-244
Author(s):  
Adel N. Boules
Keyword(s):  
Topological Spaces ◽  
Hausdorff Spaces ◽  
Locally Compact ◽  
Compact Spaces ◽  
Section 8 ◽  
Finite Products ◽  
The One ◽  
One Point Compactification ◽  
Locally Compact Spaces ◽  
Metrization Theorem

The first eight sections of this chapter constitute its core and are generally parallel to the leading sections of chapter 4. Most of the sections are brief and emphasize the nonmetric aspects of topology. Among the topics treated are normality, regularity, and second countability. The proof of Tychonoff’s theorem for finite products appears in section 8. The section on locally compact spaces is the transition between the core of the chapter and the more advanced sections on metrization, compactification, and the product of infinitely many spaces. The highlights include the one-point compactification, the Urysohn metrization theorem, and Tychonoff’s theorem. Little subsequent material is based on the last three sections. At various points in the book, it is explained how results stated for the metric case can be extended to topological spaces, especially locally compact Hausdorff spaces. Some such results are developed in the exercises.

scholarly journals Semigroups of continuous selfmaps for which Green's and ℐ relations coincide

1979 ◽  
Vol 20 (1) ◽  
pp. 25-28 ◽  
Author(s):  
K. D. Magill
Keyword(s):  
Topological Space ◽  
Metric Spaces ◽  
Transformation Semigroup ◽  
Doctoral Dissertation ◽  
Discrete Space ◽  
Full Transformation ◽  
Full Transformation Semigroup ◽  
The One ◽  
One Point Compactification ◽  
Countably Infinite

For algebraic terms which are not defined, one may consult [2]. The symbol S(X) denotes the semigroup, under composition, of all continuous selfmaps of the topological space X. When X is discrete, S(X) is simply the full transformation semigroup on the set X. It has long been known that Green's relations and ℐ coincide for [2, p. 52] and F. A. Cezus has shown in his doctoral dissertation [1, p. 34] that and ℐ also coincide for S(X) when X is the one-point compactification of the countably infinite discrete space. Our main purpose here is to point out the fact that among the 0-dimensional metric spaces, Cezus discovered the only nondiscrete space X with the property that and ℐ coincide on the semigroup S(X). Because of a result in a previous paper [6] by S. Subbiah and the author, this property (for 0-dimensional metric spaces) is in turn equivalent to the semigroup being regular. We gather all this together in the following

An additivity formula for the strict global dimension of C(Ω)

2014 ◽  
Vol 12 (3) ◽  
Author(s):  
Seytek Tabaldyev
Keyword(s):  
Topological Space ◽  
Tensor Product ◽  
Banach Algebra ◽  
Continuous Functions ◽  
Global Dimension ◽  
Large Cardinality ◽  
The One ◽  
One Point Compactification

AbstractLet A be a unital strict Banach algebra, and let K + be the one-point compactification of a discrete topological space K. Denote by the weak tensor product of the algebra A and C(K +), the algebra of continuous functions on K +. We prove that if K has sufficiently large cardinality (depending on A), then the strict global dimension is equal to .

scholarly journals BOUNDING THE MINIMUM ORDER OF Aπ-BASE

2011 ◽  
Vol 83 (2) ◽  
pp. 321-328
Author(s):  
MARÍA MUÑOZ
Keyword(s):  
Topological Space ◽  
Topological Spaces ◽  
Hausdorff Spaces ◽  
Minimum Order ◽  
Compact Spaces ◽  
Cardinal Invariants ◽  
Open Set ◽  
Open Questions ◽  
The Family ◽  
The One

AbstractLetXbe a topological space. A family ℬ of nonempty open sets inXis called aπ-base ofXif for each open setUinXthere existsB∈ℬ such thatB⊂U. The order of aπ-base ℬ at a pointxis the cardinality of the family ℬx={B∈ℬ:x∈B} and the order of theπ-base ℬ is the supremum of the orders of ℬ at each pointx∈X. A classical theorem of Shapirovskiĭ [‘Special types of embeddings in Tychonoff cubes’, in:Subspaces of Σ-Products and Cardinal Invariants, Topology, Coll. Math. Soc. J. Bolyai, 23 (North-Holland, Amsterdam, 1980), pp. 1055–1086; ‘Cardinal invariants in compact Hausdorff spaces’,Amer. Math. Soc. Transl.134(1987), 93–118] establishes that the minimum order of aπ-base is bounded by the tightness of the space when the space is compact. Since then, there have been many attempts at improving the result. Finally, in [‘The projectiveπ-character bounds the order of aπ-base’,Proc. Amer. Math. Soc.136(2008), 2979–2984], Juhász and Szentmiklóssy proved that the minimum order of aπ-base is bounded by the ‘projectiveπ-character’ of the space for any topological space (not only for compact spaces), improving Shapirovskiĭ’s theorem. The projectiveπ-character is in some sense an ‘external’ cardinal function. Our purpose in this paper is, on the one hand, to give bounds of the projectiveπ-character using ‘internal’ topological properties of the subspaces on compact spaces. On the other hand, we give a bound on the minimum order of aπ-base using other cardinal functions in the frame of general topological spaces. Open questions are posed.

scholarly journals One-point compactification on convergence spaces

1994 ◽  
Vol 17 (2) ◽  
pp. 277-282
Author(s):  
Shing S. So
Keyword(s):  
Topological Spaces ◽  
Convergence Space ◽  
Locally Compact ◽  
Convergence Spaces ◽  
The One ◽  
One Point Compactification

A convergence space is a set together with a notion of convergence of nets. It is well known how the one-point compactification can be constructed on noncompact, locally compact topological spaces. In this paper, we discuss the construction of the one-point compactification on noncompact convergence spaces and some of the properties of the one-point compactification of convergence spaces are also discussed.

scholarly journals Antisymmetry of the Stochastical Order on all Ordered Topological Spaces

2019 ◽  
Vol 7 (1) ◽  
pp. 250-252 ◽  
Author(s):  
Tobias Fritz
Keyword(s):  
Topological Space ◽  
General Result ◽  
Elementary Proof ◽  
Short Note ◽  
Stochastic Order ◽  
Topological Spaces ◽  
Probability Measures ◽  
Ordered Topological Space ◽  
Special Cases

Abstract In this short note, we prove that the stochastic order of Radon probability measures on any ordered topological space is antisymmetric. This has been known before in various special cases. We give a simple and elementary proof of the general result.

On topological quotient maps preserved by pullbacks or products

1970 ◽  
Vol 67 (3) ◽  
pp. 553-558 ◽  
Author(s):  
B. J. Day ◽  
G. M. Kelly
Keyword(s):  
Topological Space ◽  
Topological Spaces ◽  
Quotient Maps ◽  
Continuous Maps

We are concerned with the category of topological spaces and continuous maps. A surjection f: X → Y in this category is called a quotient map if G is open in Y whenever f−1G is open in X. Our purpose is to answer the following three questions:Question 1. For which continuous surjections f: X → Y is every pullback of f a quotient map?Question 2. For which continuous surjections f: X → Y is f × lz: X × Z → Y × Z a quotient map for every topological space Z? (These include all those f answering to Question 1, since f × lz is the pullback of f by the projection map Y ×Z → Y.)Question 3. For which topological spaces Z is f × 1Z: X × Z → Y × Z a qiptoent map for every quotient map f?

scholarly journals Separation Axioms in Intuitionistic Fuzzy Topological Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-7
Author(s):  
Amit Kumar Singh ◽  
Rekha Srivastava
Keyword(s):  
Topological Space ◽  
Left Adjoint ◽  
Topological Spaces ◽  
Intuitionistic Fuzzy ◽  
Separation Axioms ◽  
Fuzzy Topological Space ◽  
Fuzzy Topological Spaces

In this paper we have studied separation axiomsTi,i=0,1,2in an intuitionistic fuzzy topological space introduced by Coker. We also show the existence of functorsℬ:IF-Top→BF-Topand𝒟:BF-Top→IF-Topand observe that𝒟is left adjoint toℬ.

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