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<p>A space <i>X</i> is od-Menger if it satisfies <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\({\textsf{U}_{\textsf{fin}}}\bigl ({\Delta _X},{\mathcal {O}_X}\bigr )\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="sans-serif">U</mi> <mi mathvariant="sans-serif">fin</mi> </msub> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">(</mo> </mrow> <msub> <mi mathvariant="normal">Δ</mi> <mi>X</mi> </msub> <mo>,</mo> <msub> <mi mathvariant="script">O</mi> <mi>X</mi> </msub> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\mathcal {O}_X,\Delta _X\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="script">O</mi> <mi>X</mi> </msub> <mo>,</mo> <msub> <mi mathvariant="normal">Δ</mi> <mi>X</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> are the collection of covers of <i>X</i> by respectively open subsets and open dense subsets. We show that under <b>CH</b>, there is a refinement of the usual topology on a subset of the reals which yields a hereditarily Lindelöf, od-Menger, non-Menger, 0-dimensional, first countable space. We also investigate the properties of spaces which are od-Menger but not Menger.</p>

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On (non-Menger) spaces whose closed nowhere dense subsets are Menger

  • 期刊论文

  • 发表日期：

    2026年5月20日

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  • Mathieu Baillif,
  • Santi Spadaro

摘要

A space X is od-Menger if it satisfies \({\textsf{U}_{\textsf{fin}}}\bigl ({\Delta _X},{\mathcal {O}_X}\bigr )\) U fin ( Δ X , O X ) , where \(\mathcal {O}_X,\Delta _X\) O X , Δ X are the collection of covers of X by respectively open subsets and open dense subsets. We show that under CH, there is a refinement of the usual topology on a subset of the reals which yields a hereditarily Lindelöf, od-Menger, non-Menger, 0-dimensional, first countable space. We also investigate the properties of spaces which are od-Menger but not Menger.