Relations of topologies as tools of (bi)topological applications
type: Item
instance of: scholarly article
title: Relations of topologies as tools of (bi)topological applications
zbMATH Open document ID: 1333.54007
author: Badri P. Dvalishvili
published in: Georgian Mathematical Journal
publication date: 2015-12-07T00:00:00Z
review text: For a topological space \((X,\tau_1)\), \(A\subset X\) and \(A\not\in\tau_1\), the topology \(\tau_1(A)= \{U\cup(V\cap A): U,V\in\tau_1\}\) is said to be a simple extension of the topology \(\tau_1\); the bitopological space \((X,\tau_1< \tau_1(A))\) is of interest in the paper. For two topologies \(\tau_1\) and \(\tau_2\) on a set \(X\), \(\tau_1\text{\,imp\,}\tau_2\) iff (i) \(\tau_1\subset\tau_2\), \(\tau_1\neq\tau_2\); (ii) if \(\tau_1\subset\tau\subset\tau_2\) (for some topology \(\tau\) on \(X\)), then \(\tau_1=\tau\) or \(\tau_2=\tau\) and further, \(\tau_1\) and \(\tau_2\) are called adjacent iff \(\tau_1\text{\,imp\,}\tau_2\) or \(\tau_2\text{\,imp\,}\tau_1\). Primarily, the paper is interested in studying some properties of simple extensions and adjacent topologies; the interesting problem of passage of some important properties from \(\tau_1\) to \(\tau_1(A)\) is explored. Among the properties, as mentioned, the Blumberg property, pseudo-completeness, connectedness, submaximality, to be a Baire space, to be an \(H\)-closed space, to be an \(HD\)-space, to be an \(eD\)-space and to be a nodec space are worth mentioning. The concepts of \(HD\)-space and \(eD\)-space are introduced by the author; if, for each neighbourhood assignment \(\phi:X\to\tau\), there exists a closed discrete subset \(F\) of \((X,\tau)\) such that \(\bigcup \{\tau\text{\,cl\,} \phi(x): x\in F\}= X\), then \((X,\tau)\) is called an \(HD\)-space; if further, for each open neighborhood assignment \(\phi\), there are an open neighborhood assignment \(\psi:X\to\tau\) and a closed discrete subset \(F\) of \((X,\tau)\) such that \(\phi(x)\subset\psi(x)\), for each \(x\in X\) and \(\bigcup\{\psi(x): x\in F\}= X\), then \((X,\tau)\) is called an \(eD\)-space. It is proved that every countably compact \(HD\)-space is \(H\)-closed and \((X,\tau')\) is an \(eD\)-space for each topology \(\tau'\supseteq\tau\) on \(X\) when \((X,\tau)\) is a \(D\)-space. For a connected \(T_1\)-space \((X,\tau_1)\)…
Mathematics Subject Classification ID: 54A10
Mathematics Subject Classification ID: 54E55
zbMATH DE Number: 6516787
zbMATH Keywords: \(S\)-relation
zbMATH Keywords: coupling of topologies
zbMATH Keywords: nearness of topologies
zbMATH Keywords: simple extension
zbMATH Keywords: adjacent topologies
zbMATH Keywords: weak Lindelöf number
zbMATH Keywords: \(HD\)-space
zbMATH Keywords: \(eD\)-space
zbMATH Keywords: \((i,j)\)-submaximal space
zbMATH Keywords: \((i,j)\)-singular set
MaRDI profile type: Publication
full work available at URL: gmj-2015-0047
OpenAlex ID: W2408442099
DOI: 10.1515/GMJ-2015-0047
description: scientific article; zbMATH DE number 6516787