Abstract <p> A set <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11006_2025_3029_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">\(X\)</EquationSource> </InlineEquation> with a distance function <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11006_2025_3029_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="91" /> </InlineMediaObject> <EquationSource Format="TEX">\(\rho\colon X^2\to\mathbb R_+\)</EquationSource> </InlineEquation> satisfying the identity axiom, that is, such that <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11006_2025_3029_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="79" /> </InlineMediaObject> <EquationSource Format="TEX">\(\rho(x,y)=0\)</EquationSource> </InlineEquation> if and only if <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11006_2025_3029_Article_IEq4.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="TEX">\(x=y\)</EquationSource> </InlineEquation>, is considered. The function <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11006_2025_3029_Article_IEq5.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">\(\rho\)</EquationSource> </InlineEquation> determines a topology on <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11006_2025_3029_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">\(X\)</EquationSource> </InlineEquation>; a set <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11006_2025_3029_Article_IEq7.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="55" /> </InlineMediaObject> <EquationSource Format="TEX">\(U\subset X\)</EquationSource> </InlineEquation> belongs to this topology if and only if, for each <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11006_2025_3029_Article_IEq8.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="47" /> </InlineMediaObject> <EquationSource Format="TEX">\(u\in U\)</EquationSource> </InlineEquation>, there exists a positive <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11006_2025_3029_Article_IEq9.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">\(\delta\)</EquationSource> </InlineEquation> such that <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11006_2025_3029_Article_IEq10.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="157" /> </InlineMediaObject> <EquationSource Format="TEX">\(\{x\colon \rho(u,x)<\delta\}\subset U\)</EquationSource> </InlineEquation>. Closedness, sequential closedness, compactness, sequential compactness, and total boundedness in topological spaces thus arising are studied. </p>

Topological Properties of Spaces with a Distance

期刊论文
发表日期：

2025年2月1日

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E. S. Zhukovskiy

摘要

Abstract

A set \(X\) with a distance function \(\rho\colon X^2\to\mathbb R_+\) satisfying the identity axiom, that is, such that \(\rho(x,y)=0\) if and only if \(x=y\) , is considered. The function \(\rho\) determines a topology on \(X\) ; a set \(U\subset X\) belongs to this topology if and only if, for each \(u\in U\) , there exists a positive \(\delta\) such that \(\{x\colon \rho(u,x)<\delta\}\subset U\) . Closedness, sequential closedness, compactness, sequential compactness, and total boundedness in topological spaces thus arising are studied.