We establish that, for any Tychonoff space X, at least one of the spaces <InlineEquation ID="IEq9"> <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 <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$C_pC_p(X)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>C</mi> <mi>p</mi> </msub> <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 a dense subspace of countable pseudocharacter. Under MA, we give an example of a space X such that <InlineEquation ID="IEq11"> <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> does not have a dense subspace of countable functional tightness. We also show that there exists a compact zero-dimensional space K such that <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$C_p(K,\{0,1\})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>C</mi> <mi>p</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>K</mi> <mo>,</mo> <mrow> <mo stretchy="false">{</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">}</mo> </mrow> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> is exponentially separable while K is not a Sokolov space. For compact scattered spaces K of countable dispersion index, we show that <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$C_p(K)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>C</mi> <mi>p</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>K</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> has a dense exponentially separable subspace if and only if K is <InlineEquation ID="IEq14"> <EquationSource Format="TEX">$\omega $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ω</mi> </math></EquationSource> </InlineEquation>-monolithic. Our results provide answers to several published open questions.

For any space X, either $C_p(X)$ or $C_pC_p(X)$ is $\psi $-separable

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2026年2月1日

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J. A. Aguilar-Velázquez,
V. V. Tkachuk

摘要

We establish that, for any Tychonoff space X, at least one of the spaces $C_p(X)$ $C_{p} (X)$ and $C_pC_p(X)$ $C_{p} C_{p} (X)$ has a dense subspace of countable pseudocharacter. Under MA, we give an example of a space X such that $C_p(X)$ $C_{p} (X)$ does not have a dense subspace of countable functional tightness. We also show that there exists a compact zero-dimensional space K such that $C_p(K,\{0,1\})$ $C_{p} (K, {0, 1})$ is exponentially separable while K is not a Sokolov space. For compact scattered spaces K of countable dispersion index, we show that $C_p(K)$ $C_{p} (K)$ has a dense exponentially separable subspace if and only if K is $\omega $ $ω$ -monolithic. Our results provide answers to several published open questions.

For any space X, either \(C_p(X)\) or \(C_pC_p(X)\) is \(\psi \)-separable

摘要