Epimorphisms From S(X) onto S(Y)

1986 ◽  
Vol 38 (3) ◽  
pp. 538-551 ◽  
Author(s):  
K. D. Magill ◽  
P. R. Misra ◽  
U. B. Tewari
Keyword(s):  
Topological Space ◽  
Hausdorff Spaces ◽  
Severe Restriction ◽  
Completely Regular ◽  
Image Position

1. Introduction. In this paper, the expression topological space will always mean generated space, that is any T1 space X for whichforms a subbasis for the closed subsets of X. This is not at all a severe restriction since generated spaces include all completely regular Hausdorff spaces which contain an arc as well as all 0-dimensional Hausdorff spaces [3, pp. 198-201], [4].The symbol S(X) denotes the semigroup, under composition, of all continuous selfmaps of the topological space X. This paper really grew out of our efforts to determine all those congruences σ on S(X) such that S(X)/σ is isomorphic to S(Y) for some space Y.

Separating Closed Sets by Continuous Mappings into Developable Spaces

1981 ◽  
Vol 33 (6) ◽  
pp. 1420-1431 ◽  
Author(s):  
Harald Brandenburg
Keyword(s):  
Topological Space ◽  
Double Sequence ◽  
Completely Regular ◽  
Closed Sets ◽  
Continuous Mappings ◽  
Neighbourhood Base ◽  
Image Position ◽  
Metrizable Spaces ◽  
Metrization Theorem ◽  
Developable Spaces

A topological space X is called developable if it has a development, i.e., a sequence of open covers of X such that for each x ∈ X the collection is a neighbourhood base of x, whereThis class of spaces has turned out to be one of the most natural and useful generalizations of metrizable spaces [23]. In [4] it was shown that some well known results in metrization theory have counterparts in the theory of developable spaces (i.e., Urysohn's metrization theorem, the Nagata-Smirnov theorem, and Nagata's “double sequence theorem”). Moreover, in [3] it was pointed out that subspaces of products of developable spaces (i.e., D-completely regular spaces) can be characterized in much the same way as subspaces of products of metrizable spaces (i.e., completely regular T1-spaces).

scholarly journals Perfect Mappings and Spaces of Countable Type

1970 ◽  
Vol 22 (6) ◽  
pp. 1208-1210 ◽  
Author(s):  
J. E. Vaughan
Keyword(s):  
Topological Space ◽  
Affirmative Answer ◽  
Regular Space ◽  
Hausdorff Spaces ◽  
Completely Regular ◽  
Countable Type ◽  
Open Set

In [1, p. 41, Theorem 3.10] Arhangel'skiï proved that the perfect image of a completely regular space of countable type is of countable type, and he asked [1, p. 60, problem 4] if a similar result held for regular or Hausdorff spaces. In this paper, it is proved that the perfect image of a space of countable type is of countable type, provided that the image is Hausdorff or regular. An affirmative answer to both of Arhangel'skiï's questions follows immediately from this. Arhangel'skiï made use of the Stone-Čech compactification in the proof of his result, but the proofs below are of a different nature.Let X be a topological space and let K ⊂ X. A collection of open sets is called a base at K provided that for every open set W ⊃ K there exists such that K ⊂ U ⊂ W. Clearly, we may assume that every member of contains K.

scholarly journals Quasicompactness and functionally Hausdorff spaces

1973 ◽  
Vol 15 (3) ◽  
pp. 319-324 ◽  
Author(s):  
A. J. D'Aristotle
Keyword(s):  
Continuous Function ◽  
Topological Space ◽  
Hausdorff Spaces ◽  
Completely Regular ◽  
Separation Axioms

In [1] Van Est and Freudenthal introduced and studied several new separation axioms for a topological space. One of these was the pτsq axiom: Given distinct points p and q of X, there exists a real continuous function f on X with f(p) ≠ f(q). They observed that the pτsq axiom lies strictly between the Hausdorff and completely regular axioms, and that it neither implies nor is implied by the T3 axiom.

Continuous images of proper analytic and proper Borel spaces

1974 ◽  
Vol 75 (2) ◽  
pp. 185-191 ◽  
Author(s):  
J. E. Jayne
Keyword(s):  
Topological Space ◽  
Topological Spaces ◽  
Hausdorff Spaces ◽  
Completely Regular ◽  
Continuous Images

All topological spaces considered will be completely regular Hausdorff spaces. The word space will be used to refer to such a topological space.

Monadic binary relations and the monad systems at near-standard points

1987 ◽  
Vol 52 (3) ◽  
pp. 689-697
Author(s):  
Nader Vakil
Keyword(s):  
Topological Space ◽  
Binary Relation ◽  
Hausdorff Space ◽  
Equivalence Relations ◽  
Binary Relations ◽  
Completely Regular ◽  
General Topological Space ◽  
Image Position

AbstractLet (*X, *T) be the nonstandard extension of a Hausdorff space (X, T). After Wattenberg [6], the monad m(x) of a near-standard point x in *X is defined as m{x) = μT(st(x)). Consider the relationFrank Wattenberg in [6] and [7] investigated the possibilities of extending the domain of Rns to the whole of *X. Wattenberg's extensions of Rns were required to be equivalence relations, among other things. Because the nontrivial ways of constructing such extensions usually produce monadic relations, the said condition practically limits (to completely regular spaces) the class of spaces for which such extensions are possible. Since symmetry and transitivity are not, after all, characteristics of the kind of nearness that is obtained in a general topological space, it may be expected that if these two requirements are relaxed, then a monadic extension of Rns to *X should be possible in any topological space. A study of such extensions of Rns is the purpose of the present paper. We call a binary relation W ⊆ *X × *X an infinitesimal on *X if it is monadic and reflexive on *X. We prove, among other things, that the existence of an infinitesimal on *X that extends Rns is equivalent to the condition that the space (X, T) be regular.

scholarly journals Two semigroups of continuous relations

1977 ◽  
Vol 23 (1) ◽  
pp. 46-58 ◽  
Author(s):  
A. R. Bednarek ◽  
Eugene M. Norris
Keyword(s):  
Large Class ◽  
Algebraic Structure ◽  
Topological Spaces ◽  
Hausdorff Spaces ◽  
Completely Regular ◽  
Topological Semigroups ◽  
Stone Type

SynopsisIn this paper we define two semigroups of continuous relations on topological spaces and determine a large class of spaces for which Banach-Stone type theorems hold, i.e. spaces for which isomorphism of the semigroups implies homeomorphism of the spaces. This class includes all 0-dimensional Hausdorff spaces and all those completely regular Hausdorff spaces which contain an arc; indeed all of K. D. Magill's S*-spaces are included. Some of the algebraic structure of the semigroup of all continuous relations is elucidated and a method for producing examples of topological semigroups of relations is discussed.

Uniformity on generalized topological spaces

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Dipankar Dey ◽  
Dhananjay Mandal ◽  
Manabendra Nath Mukherjee
Keyword(s):  
Topological Space ◽  
Present Article ◽  
Design Methodology ◽  
Original Work ◽  
Generalized Topology ◽  
Topological Spaces ◽  
Intermediate Structure ◽  
Content Type ◽  
Completely Regular ◽  
Generalized Topological Space

PurposeThe present article deals with the initiation and study of a uniformity like notion, captioned μ-uniformity, in the context of a generalized topological space.Design/methodology/approachThe existence of uniformity for a completely regular topological space is well-known, and the interrelation of this structure with a proximity is also well-studied. Using this idea, a structure on generalized topological space has been developed, to establish the same type of compatibility in the corresponding frameworks.FindingsIt is proved, among other things, that a μ-uniformity on a non-empty set X always induces a generalized topology on X, which is μ-completely regular too. In the last theorem of the paper, the authors develop a relation between μ-proximity and μ-uniformity by showing that every μ-uniformity generates a μ-proximity, both giving the same generalized topology on the underlying set.Originality/valueIt is an original work influenced by the previous works that have been done on generalized topological spaces. A kind of generalization has been done in this article, that has produced an intermediate structure to the already known generalized topological spaces.

On Certain Onto Maps

1962 ◽  
Vol 14 ◽  
pp. 461-466 ◽  
Author(s):  
Isaac Namioka
Keyword(s):  
Topological Space ◽  
Closed Subspace ◽  
Dimensional Space ◽  
Extension Theorem ◽  
Normal Space ◽  
Continuous Map ◽  
Image Position

Let Δn (n > 0) denote the subset of the Euclidean (n + 1)-dimensional space defined byA subset σ of Δn is called a face if there exists a sequence 0 ≤ i1 ≤ i2 ≤ … < im ≤ n such thatand the dimension of σ is defined to be (n — m). Let denote the union of all faces of Δn of dimensions less than n. A topological space Y is called solid if any continuous map on a closed subspace A of a normal space X into Y can be extended to a map on X into Y. By Tietz's extension theorem, each face of Δn is solid. The present paper is concerned with a generalization of the following theorem which seems well known.

A Characterization of Universal Loeb Measurability for Completely Regular Hausdorff Spaces

1992 ◽  
Vol 44 (4) ◽  
pp. 673-690 ◽  
Author(s):  
J. M. Aldaz
Keyword(s):  
Hausdorff Spaces ◽  
Completely Regular ◽  
Standard Part ◽  
Full Structure

AbstractIn this paper it is shown that the construction of measures on standard spaces via Loeb measures and the standard part map does not depend on the full structure of the internal algebra being used. A characterization of universal Loeb measurability is given for completely regular Hausdorff spaces, and the behavior of this property under various topological operations is investigated.

Diamond Principles, Ideals and the Normal Moore Space Problem

1981 ◽  
Vol 33 (2) ◽  
pp. 282-296 ◽  
Author(s):  
Alan D. Taylor
Keyword(s):  
Topological Space ◽  
Pairwise Disjoint ◽  
Space Problem ◽  
Moore Space ◽  
Image Position

If is a topological space then a sequence (Cα:α < λ) of subsets of is said to be normalized if for every H ⊆ λ there exist disjoint open sets and such thatThe sequence (Cα:α < λ) is said to be separated if there exists a sequence of pairwise disjoint open sets such that for each α < λ. As is customary, we adopt the convention that all sequences (Cα:α < λ) considered are assumed to be relatively discrete as defined in [18, p. 21]: if x ∈ Cα then there exists a neighborhood about x that intersects no Cβ for β ≠ α.

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