We present a negative answer to Tkachuk's question if there is a Tychonoff space X such that <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$C_p(X)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>C</mi> <mi>p</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> has the Fréchet–Urysohn property and Player II has a winning strategy in <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$CD(C_p(X))$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>C</mi> <mi>D</mi> <mo stretchy="false">(</mo> <msub> <mi>C</mi> <mi>p</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, as well as a positive one to Clontz and Holshouser’s question whether there is a point-picking game on <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$C_p(X)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>C</mi> <mi>p</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> which characterizes when X is not a Rothberger space. In reality, we prove a stronger statement: that there is a game on <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$C_p(X)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>C</mi> <mi>p</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> which is equivalent to the Rothberger game on X. Moreover, we prove the equivalence between pairs of games, one of which played on <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$C_p(X)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>C</mi> <mi>p</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and one on X, by providing a two-way “translation” of strategies.

On the connection between games on a space and on its $C_p$-space

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发表日期：

2025年12月1日

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L. Chiozini

摘要

We present a negative answer to Tkachuk's question if there is a Tychonoff space X such that $C_p(X)$ $C_{p} (X)$ has the Fréchet–Urysohn property and Player II has a winning strategy in $CD(C_p(X))$ $C D (C_{p} (X))$ , as well as a positive one to Clontz and Holshouser’s question whether there is a point-picking game on $C_p(X)$ $C_{p} (X)$ which characterizes when X is not a Rothberger space. In reality, we prove a stronger statement: that there is a game on $C_p(X)$ $C_{p} (X)$ which is equivalent to the Rothberger game on X. Moreover, we prove the equivalence between pairs of games, one of which played on $C_p(X)$ $C_{p} (X)$ and one on X, by providing a two-way “translation” of strategies.

On the connection between games on a space and on its \(C_p\)-space

摘要