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This chapter deals with K-analytic and analytic spaces \(C_p(X)\) . Some results due to Talagrand, Tkachuk, Velichko, and Canela are presented. A remarkable theorem of Christensen stating that a metrizable and separable space X is \(\sigma \) -compact if and only if \(C_p(X)\) is analytic is proved. We show that the analyticity of \(C_p(X)\) for any X implies that X is \(\sigma \) -compact (Calbrix’s theorem). We show that \(C_p(X)\) is K-analytic-framed in \(\mathbb {R}^X\) if and only if \(C_p(X)\) admits a bounded resolution. We also gather several equivalent conditions for spaces \(C_p(X)\) to be Lindelöf spaces over locally compact groups X.

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K-Analytic and Analytic Spaces \(C_{p}(X)\)

  • 章节

  • 发表日期：

    2025年

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  • Jerzy Kąkol,
  • Wiesław Kubiś,
  • Manuel López-Pellicer,
  • Damian Sobota

摘要

This chapter deals with K-analytic and analytic spaces \(C_p(X)\) . Some results due to Talagrand, Tkachuk, Velichko, and Canela are presented. A remarkable theorem of Christensen stating that a metrizable and separable space X is \(\sigma \) -compact if and only if \(C_p(X)\) is analytic is proved. We show that the analyticity of \(C_p(X)\) for any X implies that X is \(\sigma \) -compact (Calbrix’s theorem). We show that \(C_p(X)\) is K-analytic-framed in \(\mathbb {R}^X\) if and only if \(C_p(X)\) admits a bounded resolution. We also gather several equivalent conditions for spaces \(C_p(X)\) to be Lindelöf spaces over locally compact groups X.