Every Hausdorff Compactification of a Locally Compact Separable Space is a Ga Compactification

In [4], De Groot and Aarts constructed Hausdorff compactifications of topological spaces to obtain a new intrinsic characterization of complete regularity. These compactifications were called GA compactifications in [5] and [7]. A characterization of complete regularity was earlier given by Frink [3], by means of Wallman compactifications, a method which led to the intriguing problem of whether every Hausdorff compactification is a Wallman compactification.

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On Hausdorff compactifications of non-locally compact spaces

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/s0161171279000375 ◽

1979 ◽

Vol 2 (3) ◽

pp. 481-486

Author(s):

James Hatzenbuhler ◽

Don A. Mattson

Keyword(s):

Hausdorff Space ◽

Locally Compact ◽

Completely Regular ◽

Compact Spaces ◽

Hausdorff Compactifications ◽

Set Of Points ◽

Locally Compact Spaces

LetXbe a completely regular, Hausdorff space and letRbe the set of points inXwhich do not possess compact neighborhoods. AssumeRis compact. IfXhas a compactification with a countable remainder, then so does the quotientX/R, and a countable compactificatlon ofX/Rimplies one forX−R. A characterization of whenX/Rhas a compactification with a countable remainder is obtained. Examples show that the above implications cannot be reversed.

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On product of spaces of quasicomponents

Filomat ◽

10.2298/fil1720307m ◽

2017 ◽

Vol 31 (20) ◽

pp. 6307-6311

Author(s):

Gjorgji Markoski ◽

Abdulla Buklla

Keyword(s):

Product Space ◽

Continuous Functions ◽

Topological Spaces ◽

Topological Properties ◽

Locally Compact

We use a characterization of quasicomponents by continuous functions to obtain the well known theorem which states that product of quasicomponents Qx,Qy of topological spaces X,Y, respectively, gives quasicomponent in the product space X x Y. If spaces X,Y are locally-compact, paracompact and Haussdorf, then we prove that the space of quasicomponents of the product Q(XxY) is homeomorphic with the product space Q(X) x Q(Y), so these two spaces have the same topological properties.

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A Hahn-Banach Theorem in Subbase Convexity Theory

Canadian Journal of Mathematics ◽

10.4153/cjm-1980-061-x ◽

1980 ◽

Vol 32 (4) ◽

pp. 804-820 ◽

Cited By ~ 1

Author(s):

M. van de Vel

Keyword(s):

Main Tool ◽

Complete Regularity ◽

Completely Regular ◽

Banach Theorem ◽

Convexity Theory ◽

Hausdorff Compactifications

In the last fifteen years, topology has shown up with an increasing interest in the use of closed subbases. Starting from Frink's internal characterization of complete regularity (Frink [6]), DeGroot and Aarts used closed subbases to obtain Hausdorff compactifications of completely regular spaces, thus giving a characterization of the latter in terms of their subbases [1]. The main tool of that paper is the notion of a linked system, which naturally leads to the notions of supercompactness and superextensions [7]. After 1970, these two topics developed to indepedennt theories, with several deep results available at this moment. Most results up to 1976 are summarized in [12].

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An Intrinsic Characterization of Five Points in a CAT(0) Space

Analysis and Geometry in Metric Spaces ◽

10.1515/agms-2020-0111 ◽

2020 ◽

Vol 8 (1) ◽

pp. 114-165

Author(s):

Tetsu Toyoda

Keyword(s):

Metric Space ◽

Isometric Embedding ◽

Metric Spaces ◽

Necessary Conditions ◽

New Approach ◽

Intrinsic Characterization ◽

New Family

AbstractGromov (2001) and Sturm (2003) proved that any four points in a CAT(0) space satisfy a certain family of inequalities. We call those inequalities the ⊠-inequalities, following the notation used by Gromov. In this paper, we prove that a metric space X containing at most five points admits an isometric embedding into a CAT(0) space if and only if any four points in X satisfy the ⊠-inequalities. To prove this, we introduce a new family of necessary conditions for a metric space to admit an isometric embedding into a CAT(0) space by modifying and generalizing Gromov’s cycle conditions. Furthermore, we prove that if a metric space satisfies all those necessary conditions, then it admits an isometric embedding into a CAT(0) space. This work presents a new approach to characterizing those metric spaces that admit an isometric embedding into a CAT(0) space.

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Homological epimorphisms, homotopy epimorphisms and acyclic maps

Forum Mathematicum ◽

10.1515/forum-2019-0249 ◽

2020 ◽

Vol 32 (6) ◽

pp. 1395-1406

Author(s):

Joseph Chuang ◽

Andrey Lazarev

Keyword(s):

Wide Class ◽

Topological Spaces ◽

Loop Spaces ◽

Graded Algebras ◽

Differential Graded Algebras ◽

Algebraic Description ◽

Acyclic Map ◽

Induced Maps

AbstractWe show that the notions of homotopy epimorphism and homological epimorphism in the category of differential graded algebras are equivalent. As an application we obtain a characterization of acyclic maps of topological spaces in terms of induced maps of their chain algebras of based loop spaces. In the case of a universal acyclic map we obtain, for a wide class of spaces, an explicit algebraic description for these induced maps in terms of derived localization.

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On Homology and Cohomology Groups of Remainders

Georgian Mathematical Journal ◽

10.1515/gmj.2004.613 ◽

2004 ◽

Vol 11 (4) ◽

pp. 613-633

Author(s):

V. Baladze ◽

L. Turmanidze

Keyword(s):

Uniform Space ◽

Cohomology Groups ◽

Uniform Spaces ◽

Intrinsic Characterization

Abstract Border homology and cohomology groups of pairs of uniform spaces are defined and studied. These groups give an intrinsic characterization of Čech type homology and cohomology groups of the remainder of a uniform space.

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Scaling sets and generalized scaling sets on Cantor dyadic group

International Journal of Wavelets Multiresolution and Information Processing ◽

10.1142/s0219691320500198 ◽

2020 ◽

Vol 18 (04) ◽

pp. 2050019

Author(s):

Prasadini Mahapatra ◽

Divya Singh

Keyword(s):

Abelian Groups ◽

Real Case ◽

Locally Compact ◽

Wavelet Sets ◽

Locally Compact Abelian Groups ◽

Dyadic Group ◽

Compact Abelian Groups ◽

Cantor Dyadic Group

Scaling and generalized scaling sets determine wavelet sets and hence wavelets. In real case, wavelet sets were proved to be an important tool for the construction of MRA as well as non-MRA wavelets. However, any result related to scaling/generalized scaling sets is not available in case of locally compact abelian groups. This paper gives a characterization of scaling sets and its generalized version along with relevant examples in dual Cantor dyadic group [Formula: see text]. These results can further be generalized to arbitrary locally compact abelian groups.

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