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This chapter deals with topological (vector) spaces satisfying some sequential conditions. We study Fréchet–Urysohn spaces (i.e., spaces E such that for each \(A\subset E\) and each \(\mathrm {x}\in \mathrm {A}^{-}\) there exists a sequence in A converging to x). The main result states that every sequentially complete Fréchet–Urysohn lcs is a Baire space. Since every infinite-dimensional Montel (DF)-space E is non-metrizable and sequential, the following question arises: Is every Fréchet–Urysohn space in the class G metrizable?

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Fréchet–Urysohn Spaces and Groups

  • 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 topological (vector) spaces satisfying some sequential conditions. We study Fréchet–Urysohn spaces (i.e., spaces E such that for each \(A\subset E\) and each \(\mathrm {x}\in \mathrm {A}^{-}\) there exists a sequence in A converging to x). The main result states that every sequentially complete Fréchet–Urysohn lcs is a Baire space. Since every infinite-dimensional Montel (DF)-space E is non-metrizable and sequential, the following question arises: Is every Fréchet–Urysohn space in the class G metrizable?