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<p>We study (strong) first countability of locally solid convergence structures on Archimedean vector lattices. Among other results, we characterise those vector lattices for which relatively uniform-, order-, and <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43034_2025_433_Article_IEq1.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">\(\sigma\)</EquationSource> <EquationSource Format="MATHML"><math> <mi>σ</mi> </math></EquationSource> </InlineEquation>-order convergence, respectively, is (strongly) first countable. The implications for the validity of sequential arguments in the contexts of these convergence structures are pointed out.</p>

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Countability conditions in locally solid convergence spaces

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年9月15日

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  • Eugene Bilokopytov,
  • Viktor Bohdanskyi,
  • Jan Harm van der Walt

摘要

We study (strong) first countability of locally solid convergence structures on Archimedean vector lattices. Among other results, we characterise those vector lattices for which relatively uniform-, order-, and \(\sigma\) σ -order convergence, respectively, is (strongly) first countable. The implications for the validity of sequential arguments in the contexts of these convergence structures are pointed out.