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In this chapter, we prove that an lcs in the class \(\mathfrak {G}\) is metrizable if and only if E is b-Baire-like if and only if E is Fréchet–Urysohn. Consequently, no proper (LB)-space is Fréchet–Urysohn. We prove that if a (DF)- or (LM)-space E is sequential, then E is either metrizable or Montel (DF). We distinguish a variant of the property \(C_{3}\) (due to Webb), called property \(\mathrm {C}3\) – (i.e., sequential closure of any vector subspace is sequentially closed), and characterize both (DF)-spaces and (LF)-spaces with the property C3– as being of the form M, \(\phi \) , or \(M\times \phi \) , where M is metrizable.

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Sequential Properties in the Class \(\mathfrak {G}\)

  • 章节

  • 发表日期：

    2025年

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

摘要

In this chapter, we prove that an lcs in the class \(\mathfrak {G}\) is metrizable if and only if E is b-Baire-like if and only if E is Fréchet–Urysohn. Consequently, no proper (LB)-space is Fréchet–Urysohn. We prove that if a (DF)- or (LM)-space E is sequential, then E is either metrizable or Montel (DF). We distinguish a variant of the property \(C_{3}\) (due to Webb), called property \(\mathrm {C}3\) – (i.e., sequential closure of any vector subspace is sequentially closed), and characterize both (DF)-spaces and (LF)-spaces with the property C3– as being of the form M, \(\phi \) , or \(M\times \phi \) , where M is metrizable.