Abstract <p>A continuous map <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11982_2025_10465_Article_IEq1.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="53" /> </InlineMediaObject> <EquationSource Format="TEX">\(X\mathop \to \limits^f Y\)</EquationSource>  </InlineEquation> and its extension <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11982_2025_10465_Article_IEq2.gif" Format="GIF" Height="27" Rendition="HTML" Resolution="72" Type="Linedraw" Width="115" /> </InlineMediaObject> <EquationSource Format="TEX">\({{\exp }_{\tau }}X\mathop \to \limits^{\bar {f}} {{\exp }_{\tau }}Y\)</EquationSource>  </InlineEquation> are considered (<InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11982_2025_10465_Article_IEq3.gif" Format="GIF" Height="18" Rendition="HTML" Resolution="72" Type="Linedraw" Width="50" /> </InlineMediaObject> <EquationSource Format="TEX">\({{\exp }_{\tau }}X\)</EquationSource>  </InlineEquation> is the hyperspace (endowed with a topology <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11982_2025_10465_Article_IEq4.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="11" /> </InlineMediaObject> <EquationSource Format="TEX">\(\tau \)</EquationSource>  </InlineEquation>) of the topological space <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11982_2025_10465_Article_IEq5.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">\(X\)</EquationSource>  </InlineEquation>, <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11982_2025_10465_Article_IEq6.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="113" /> </InlineMediaObject> <EquationSource Format="TEX">\(\bar {f}(F) = [f(F{{)]}_{Y}}\)</EquationSource>  </InlineEquation> (the closure of a set <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11982_2025_10465_Article_IEq7.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="38" /> </InlineMediaObject> <EquationSource Format="TEX">\(f(F)\)</EquationSource>  </InlineEquation> in the space <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11982_2025_10465_Article_IEq8.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">\(Y\)</EquationSource>  </InlineEquation>)). A necessary and sufficient condition (a modification of the Harris (WO) condition) of continuity of the map <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11982_2025_10465_Article_IEq9.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">\(\bar {f}\)</EquationSource>  </InlineEquation> in the cases when <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11982_2025_10465_Article_IEq10.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="55" /> </InlineMediaObject> <EquationSource Format="TEX">\(\tau = {{\tau }_{{{\text{LF}}}}}\)</EquationSource>  </InlineEquation> (the locally finite topology) and <InlineEquation ID="IEq11"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11982_2025_10465_Article_IEq11.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="48" /> </InlineMediaObject> <EquationSource Format="TEX">\(\tau = {{\tau }_{{\text{F}}}}\)</EquationSource>  </InlineEquation> (the Fell topology) is found. When <InlineEquation ID="IEq12"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11982_2025_10465_Article_IEq12.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">\(X\)</EquationSource>  </InlineEquation> and <InlineEquation ID="IEq13"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11982_2025_10465_Article_IEq13.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">\(Y\)</EquationSource>  </InlineEquation> are metrizable spaces, the topology <InlineEquation ID="IEq14"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11982_2025_10465_Article_IEq14.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="22" /> </InlineMediaObject> <EquationSource Format="TEX">\({{\tau }_{{\inf }}}\)</EquationSource>  </InlineEquation>, as the infimum of all Hausdorff metric topologies, is considered. A sufficient condition (the TUC condition) of continuity of the map <InlineEquation ID="IEq15"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11982_2025_10465_Article_IEq15.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">\(\bar {f}\)</EquationSource>  </InlineEquation> in the case when <InlineEquation ID="IEq16"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11982_2025_10465_Article_IEq16.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="54" /> </InlineMediaObject> <EquationSource Format="TEX">\(\tau = {{\tau }_{{\inf }}}\)</EquationSource>  </InlineEquation> is found. It is also shown that this condition is necessary when the space <InlineEquation ID="IEq17"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11982_2025_10465_Article_IEq17.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">\(Y\)</EquationSource>  </InlineEquation> is locally compact and second countable. The results are commented from the point of view of the category theory.</p>

On the Functor Properties of Some Hyperspace Topologies

期刊论文
发表日期：

2025年2月1日

查看全文

A. S. Bedritskiy,
V. L. Timokhovich

摘要

Abstract

A continuous map \(X\mathop \to \limits^f Y\) and its extension \({{\exp }_{\tau }}X\mathop \to \limits^{\bar {f}} {{\exp }_{\tau }}Y\) are considered ( \({{\exp }_{\tau }}X\) is the hyperspace (endowed with a topology \(\tau \) ) of the topological space \(X\) , \(\bar {f}(F) = [f(F{{)]}_{Y}}\) (the closure of a set \(f(F)\) in the space \(Y\) )). A necessary and sufficient condition (a modification of the Harris (WO) condition) of continuity of the map \(\bar {f}\) in the cases when \(\tau = {{\tau }_{{{\text{LF}}}}}\) (the locally finite topology) and \(\tau = {{\tau }_{{\text{F}}}}\) (the Fell topology) is found. When \(X\) and \(Y\) are metrizable spaces, the topology \({{\tau }_{{\inf }}}\) , as the infimum of all Hausdorff metric topologies, is considered. A sufficient condition (the TUC condition) of continuity of the map \(\bar {f}\) in the case when \(\tau = {{\tau }_{{\inf }}}\) is found. It is also shown that this condition is necessary when the space \(Y\) is locally compact and second countable. The results are commented from the point of view of the category theory.