A Note on Extending Locally Finite Collections

1976 ◽  
Vol 19 (1) ◽  
pp. 117-119
Author(s):  
H. L. Shapiro ◽  
F. A. Smith
Keyword(s):  
Topological Space ◽  
Set Cover ◽  
Topological Spaces ◽  
Locally Finite ◽  
Whole Space

Recently there has been a great deal of interest in extending refinements of locally finite and point finite collections on subsets of certain topological spaces. In particular the first named author showed that a subset S of a topological space X is P-embedded in X if and only if every locally finite cozero-set cover on S has a refinement that can be extended to a locally finite cozero-set cover of X. Since then many authors have studied similar types of embeddings (see [1], [2], [3], [4], [6], [8], [9], [10], [11], and [12]). Since the above characterization of P-embedding is equivalent to extending continuous pseudometrics from the subspace S up to the whole space X, it is natural to wonder when can a locally finite or a point finite open or cozero-set cover on S be extended to a locally finite or point-finite open or cozero-set cover on X.

scholarly journals Orderability of topological spaces by continuous preferences

2001 ◽  
Vol 27 (8) ◽  
pp. 505-512 ◽  
Author(s):  
José Carlos Rodríguez Alcantud
Keyword(s):  
Topological Space ◽  
Topological Spaces ◽  
Topological Properties ◽  
Linear Orders ◽  
Mathematical Economics

We extend van Dalen and Wattel's (1973) characterization of orderable spaces and their subspaces by obtaining analogous results for two larger classes of topological spaces. This type of spaces are defined by considering preferences instead of linear orders in the former definitions, and possess topological properties similar to those of (totally) orderable spaces (cf. Alcantud, 1999). Our study provides particular consequences of relevance in mathematical economics; in particular, a condition equivalent to the existence of a continuous preference on a topological space is obtained.

More on Extending Continuous Pseudometrics

1970 ◽  
Vol 22 (5) ◽  
pp. 984-993 ◽  
Author(s):  
H. L. Shapiro
Keyword(s):  
Topological Space ◽  
Metric Space ◽  
Closed Subset ◽  
Locally Finite ◽  
Whole Space

The concept of extending to a topological space X a continuous pseudometric defined on a subspace S of X has been shown to be very useful. This problem was first studied by Hausdorff for the metric case in 1930 [9]. Hausdorff showed that a continuous metric on a closed subset of a metric space can be extended to a continuous metric on the whole space. Bing [4] and Arens [3] rediscovered this result independently. Recently, Shapiro [15] and Alo and Shapiro [1] studied various embeddings. It has been shown that extending pseudometrics can be characterized in terms of extending refinements of various types of open covers. In this paper we continue our study of extending pseudometrics. First we show that extending pseudometrics can be characterized in terms of σ-locally finite and σ-discrete covers. We then investigate when can certain types of covers be extended.

scholarly journals A characterization of pre-near-standardness in locally convex linear topological spaces

1972 ◽  
Vol 6 (1) ◽  
pp. 107-115
Author(s):  
J.J.M. Chadwick ◽  
R.W. Cross
Keyword(s):  
Topological Space ◽  
Dual Space ◽  
Linear Topological Space ◽  
Topological Spaces ◽  
Locally Convex

Let X be a locally convex linear topological space. A point z in an ultralimit enlargement of X is pre-near-standard if and only it is finite and for every equicontinuous subset S′ of the dual space X′, a point z′ belongs to *S′ ∩ μσ(X′, X) (0) only if z′ (z) is infinitesimal.

Characterization of Fuzzy δg*-Closed Sets in Fuzzy Topological Spaces

2016 ◽  
Vol 5 (2) ◽  
pp. 1-12
Author(s):  
Anahid Kamali ◽  
Hamid Reza Moradi
Keyword(s):  
Topological Space ◽  
Fuzzy Sets ◽  
Topological Spaces ◽  
Basic Properties ◽  
Closed Sets ◽  
Research Article ◽  
Classical Paper ◽  
Fuzzy Topological Space ◽  
Fuzzy Topological Spaces

The purpose of this research article is to explain the meaning of g-closed sets in fuzzy topological spaces, which is more understandable to the readers and we find some of its basic properties. The concept of fuzzy sets was introduced by Zadeh in his classical paper (1965). Thereafter many investigations have been carried out, in the general theoretical field and also in different applied areas, based on this concept. The idea of fuzzy topological space was introduced by Chang (1968). The idea is more or less a generalization of ordinary topological spaces. Different aspects of such spaces have been developed, by several investigators. This paper is also devoted to the development of the theory of fuzzy topological spaces.

scholarly journals Bicompleteness of the fine quasi-uniformity

1991 ◽  
Vol 109 (1) ◽  
pp. 167-186 ◽  
Author(s):  
Hans-Peter A. Künzi ◽  
Nathalie Ferrario
Keyword(s):  
Topological Space ◽  
Metrizable Space ◽  
Topological Spaces ◽  
Sober Space

AbstractA characterization of the topological spaces that possess a bicomplete fine quasi-uniformity is obtained. In particular we show that the fine quasi-uniformity of each sober space, of each first-countable T1-space and of each quasi-pseudo-metrizable space is bicomplete. Moreover we give examples of T1-spaces that do not admit a bicomplete quasi-uniformity.We obtain several conditions under which the semi-continuous quasi-uniformity of a topological space is bicomplete and observe that the well-monotone covering quasiuniformity of a topological space is bicomplete if and only if the space is quasi-sober.

scholarly journals Decomposition, Mapping, and Sum Theorems of ω-Paracompact Topological Spaces

Axioms ◽  
2021 ◽  
Vol 10 (4) ◽  
pp. 339
Author(s):  
Samer Al Al Ghour
Keyword(s):  
Topological Space ◽  
Topological Spaces ◽  
Locally Finite ◽  
Open Questions ◽  
Locally Countable ◽  
Sum Theorem

As a weaker form of ω-paracompactness, the notion of σ-ω-paracompactness is introduced. Furthermore, as a weaker form of σ-ω-paracompactness, the notion of feebly ω-paracompactness is introduced. It is proven hereinthat locally countable topological spaces are feebly ω-paracompact. Furthermore, it is proven hereinthat countably ω-paracompact σ-ω-paracompact topological spaces are ω-paracompact. Furthermore, it is proven hereinthat σ-ω-paracompactness is inverse invariant under perfect mappings with countable fibers, and as a result, is proven hereinthat ω-paracompactness is inverse invariant under perfect mappings with countable fibers. Furthermore, if A is a locally finite closed covering of a topological space X,τ with each A∈A being ω-paracompact and normal, then X,τ is ω-paracompact and normal, and as a corollary, a sum theorem for ω-paracompact normal topological spaces follows. Moreover, three open questions are raised.

scholarly journals The topology of θω-open sets

Filomat ◽  
2017 ◽  
Vol 31 (16) ◽  
pp. 5369-5377 ◽  
Author(s):  
Ghour Al ◽  
Bayan Irshedat
Keyword(s):  
Topological Space ◽  
Closure Operator ◽  
Sufficient Conditions ◽  
Topological Spaces ◽  
Product Theorem ◽  
Closure Operators ◽  
Separation Axiom ◽  
New Class

We define the ??-closure operator as a new topological operator. We show that ??-closure of a subset of a topological space is strictly between its usual closure and its ?-closure. Moreover, we give several sufficient conditions for the equivalence between ??-closure and usual closure operators, and between ??-closure and ?-closure operators. Also, we use the ??-closure operator to introduce ??-open sets as a new class of sets and we prove that this class of sets lies strictly between the class of open sets and the class of ?-open sets. We investigate ??-open sets, in particular, we obtain a product theorem and several mapping theorems. Moreover, we introduce ?-T2 as a new separation axiom by utilizing ?-open sets, we prove that the class of !-T2 is strictly between the class of T2 topological spaces and the class of T1 topological spaces. We study relationship between ?-T2 and ?-regularity. As main results of this paper, we give a characterization of ?-T2 via ??-closure and we give characterizations of ?-regularity via ??-closure and via ??-open sets.

A characterization of completely regular spaces

2015 ◽  
Vol 26 (03) ◽  
pp. 1550032 ◽  
Author(s):  
Richard W. M. Alves ◽  
Victor H. L. Rocha ◽  
Josiney A. Souza
Keyword(s):  
Topological Space ◽  
Regular Space ◽  
Topological Spaces ◽  
Uniform Spaces ◽  
Completely Regular ◽  
Admissible Family

This paper proves that uniform spaces and admissible spaces form the same class of topological spaces. This result characterizes a completely regular space as a topological space that admits an admissible family of open coverings. In addition, the admissible family of coverings provides an interesting methodology of studying aspects of uniformity and dynamics in completely regular spaces.

“Topologically Indexed Function Spaces and Adjoint Functors”

1982 ◽  
Vol 25 (2) ◽  
pp. 169-178
Author(s):  
S. B. Niefield
Keyword(s):  
Topological Space ◽  
Sufficient Conditions ◽  
Function Spaces ◽  
Topological Spaces ◽  
Product Topology ◽  
Adjoint Functors ◽  
Point Space ◽  
Continuous Maps ◽  
Necessary And Sufficient

AbstractLet Top denote the category of topological spaces and continuous maps. In this paper we discuss families of function spaces indexed by the elements of a topological space T, and their relationship to the characterization of right adjoints Top/S → Top/T, where S is also a topological space. After reducing the problem to the case where S is a one-point space, we describe a class of right adjoints Top → Top/T, and then show that every right adjoint Top → Top/T is isomorphic to one of this form. We conclude by giving necessary and sufficient conditions for a left adjoint Top/T → Top to be isomorphic to one of the form − XTY, where Y is a space over T, and xT denotes the fiber product with the product topology.

scholarly journals F-Rings of Continuous Functions I

1959 ◽  
Vol 11 ◽  
pp. 80-86 ◽  
Author(s):  
Barron Brainerd
Keyword(s):  
Topological Space ◽  
Banach Algebra ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Continuous Functions ◽  
The Other ◽  
Topological Spaces ◽  
Closed Sets ◽  
Real Complex

It is well known (2, 4) that the ring of all real (complex) continuous functions on a compact Hausdorff space can be characterized algebraically as a Banach algebra which satisfies certain additional intrinsic conditions. It might be expected that rings of all continuous functions on other topological spaces also have algebraic characterizations. The main purpose of this note is to discuss two such characterizations. In both cases the characterizations are given in the terms of the theory of F-brings (1). In one case a characterization is given for the ring of all (real) continuous functions on a generalized P-space, that is, a zero-dimensional topological space in which the class of open-closed sets forms a σ-algebra. A Hausdorff generalized P-space is a P-space in the terminology of (3). In the other case a theorem of Sikorski (6) is employed to give a characterization of the ring of all (real) continuous functions on an upper X1-compact P-space.

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