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Abstract <p>In this paper, we study the Baire property of the space <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(K_{1}(X,M)\)</EquationSource> <!--BMatMGU2570064Osipov-m1--> </InlineEquation>, that is, the mappings of the first functional Lebesgue class, where <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(M\)</EquationSource> <!--BMatMGU2570064Osipov-m2--> </InlineEquation> is a compact space. We study the class compact spaces for which the Baire property of the space <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(K_{1}(X,\{0,1\})\)</EquationSource> <!--BMatMGU2570064Osipov-m3--> </InlineEquation> is equivalent to the Baire property of the space <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(K_{1}(X,M)\)</EquationSource> <!--BMatMGU2570064Osipov-m4--> </InlineEquation> for any <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(M\)</EquationSource> <!--BMatMGU2570064Osipov-m5--> </InlineEquation> from this class. We prove that this class contains <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(\pi\)</EquationSource> <!--BMatMGU2570064Osipov-m6--> </InlineEquation>-monolithic compacta. In particular, a necessary and sufficient condition is obtained for the space <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(X\)</EquationSource> <!--BMatMGU2570064Osipov-m7--> </InlineEquation> under which the space <InlineEquation ID="IEq8"> <EquationSource Format="TEX">\(K_{1}(X,G)\)</EquationSource> <!--BMatMGU2570064Osipov-m8--> </InlineEquation> is Baire for any compact topological group <InlineEquation ID="IEq9"> <EquationSource Format="TEX">\(G\)</EquationSource> <!--BMatMGU2570064Osipov-m9--> </InlineEquation>.</p>

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The Baire Property for the Space of Mappings of the First Functional Lebesgue Class

  • 期刊论文

  • 发表日期：

    2025年12月1日

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  • A. V. Osipov

摘要

Abstract

In this paper, we study the Baire property of the space \(K_{1}(X,M)\) , that is, the mappings of the first functional Lebesgue class, where \(M\) is a compact space. We study the class compact spaces for which the Baire property of the space \(K_{1}(X,\{0,1\})\) is equivalent to the Baire property of the space \(K_{1}(X,M)\) for any \(M\) from this class. We prove that this class contains \(\pi\) -monolithic compacta. In particular, a necessary and sufficient condition is obtained for the space \(X\) under which the space \(K_{1}(X,G)\) is Baire for any compact topological group \(G\) .