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Abstract <p>Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(X\)</EquationSource> <!--RusMath2570076Bedritskii-m1--> </InlineEquation> and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(Y\)</EquationSource> <!--RusMath2570076Bedritskii-m2--> </InlineEquation> be metrizable spaces. A map <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(X\;\mathop \to \limits^f \;Y\)</EquationSource> <!--RusMath2570076Bedritskii-m3--> </InlineEquation> is called topologically uniformly continuous if for every admissible (i.e., consistent with topology) metric <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\rho \)</EquationSource> <!--RusMath2570076Bedritskii-m4--> </InlineEquation> on <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(X\)</EquationSource> <!--RusMath2570076Bedritskii-m5--> </InlineEquation> there exists an admissible metric <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(\sigma \)</EquationSource> <!--RusMath2570076Bedritskii-m6--> </InlineEquation> on <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(Y\)</EquationSource> <!--RusMath2570076Bedritskii-m7--> </InlineEquation> such that the map <InlineEquation ID="IEq8"> <EquationSource Format="TEX">\((X,\rho )\;\mathop \to \limits^f \;(Y,\sigma )\)</EquationSource> <!--RusMath2570076Bedritskii-m8--> </InlineEquation> is uniformly continuous for the metric spaces <InlineEquation ID="IEq9"> <EquationSource Format="TEX">\((X,\rho )\)</EquationSource> <!--RusMath2570076Bedritskii-m9--> </InlineEquation> and <InlineEquation ID="IEq10"> <EquationSource Format="TEX">\((Y,\sigma )\)</EquationSource> <!--RusMath2570076Bedritskii-m10--> </InlineEquation>. This paper analyzes such maps. The main result is that, in a certain sense, topologically uniformly continuous maps are close to perfect maps.</p>

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On Topologically Uniformly Continuous Maps

  • 期刊论文

  • 发表日期：

    2025年10月1日

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  • A. S. Bedritskiy

摘要

Abstract

Let \(X\) and \(Y\) be metrizable spaces. A map \(X\;\mathop \to \limits^f \;Y\) is called topologically uniformly continuous if for every admissible (i.e., consistent with topology) metric \(\rho \) on \(X\) there exists an admissible metric \(\sigma \) on \(Y\) such that the map \((X,\rho )\;\mathop \to \limits^f \;(Y,\sigma )\) is uniformly continuous for the metric spaces \((X,\rho )\) and \((Y,\sigma )\) . This paper analyzes such maps. The main result is that, in a certain sense, topologically uniformly continuous maps are close to perfect maps.