A well-known theorem of Noble states that each Tychonoff space X is homeomorphic to a closed subspace of a pseudocompact <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$k_\mathbb {R}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>k</mi> <mi mathvariant="double-struck">R</mi> </msub> </math></EquationSource> </InlineEquation>-space. We strengthen this result by showing that any Tychonoff space X is homeomorphic to a closed subspace of an abelian pseudocompact <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$k_\mathbb {R}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>k</mi> <mi mathvariant="double-struck">R</mi> </msub> </math></EquationSource> </InlineEquation>-group G such that <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$w(G)\le \aleph _1\cdot w(X)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>w</mi> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> <mo>≤</mo> <msub> <mi>ℵ</mi> <mn>1</mn> </msub> <mo>·</mo> <mi>w</mi> <mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, and if, in addition, X is a precompact group, then X is topologically isomorphic to a closed subgroup of G. It is constructed the first examples of pseudocompact groups <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$G_1$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>G</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$G_2$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>G</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation> (in fact, they are even countably compact and of weight <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\aleph _2$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>ℵ</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation>) such that <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$G_1$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>G</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation> is Ascoli but not a <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$k_\mathbb {R}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>k</mi> <mi mathvariant="double-struck">R</mi> </msub> </math></EquationSource> </InlineEquation>-space, and <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$G_2$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>G</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation> is a <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$k_\mathbb {R}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>k</mi> <mi mathvariant="double-struck">R</mi> </msub> </math></EquationSource> </InlineEquation>-space but not a k-space. Under <InlineEquation ID="IEq14"> <EquationSource Format="TEX">$\mathrm {MA+\lnot CH}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">MA</mi> <mo>+</mo> <mo>¬</mo> <mi mathvariant="normal">CH</mi> </mrow> </math></EquationSource> </InlineEquation>, we show that any pseudocompact group of weight <InlineEquation ID="IEq15"> <EquationSource Format="TEX">$\aleph _1$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>ℵ</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation> is Ascoli. These results are proved using topological properties of pseudocompact spaces X of weight <InlineEquation ID="IEq16"> <EquationSource Format="TEX">$\aleph _1$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>ℵ</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation> and of <InlineEquation ID="IEq17"> <EquationSource Format="TEX">$\Sigma $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Σ</mi> </math></EquationSource> </InlineEquation>-products in products of compact spaces. Being motivated by these results and the countably compact part of Noble’s theorem, it is shown by a well-known technique that each countably compact infinite group has a separable countably compact subgroup of cardinality continuum.

Closed embeddings of spaces and groups into pseudocompact $k_\mathbb {R}$-groups

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2026年2月1日

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Saak Gabriyelyan,
Jan van Mill,
Evgenii Reznichenko

摘要

A well-known theorem of Noble states that each Tychonoff space X is homeomorphic to a closed subspace of a pseudocompact $k_\mathbb {R}$ $k_{R}$ -space. We strengthen this result by showing that any Tychonoff space X is homeomorphic to a closed subspace of an abelian pseudocompact $k_\mathbb {R}$ $k_{R}$ -group G such that $w(G)\le \aleph _1\cdot w(X)$ $w (G) \leq ℵ_{1} \cdot w (X)$ , and if, in addition, X is a precompact group, then X is topologically isomorphic to a closed subgroup of G. It is constructed the first examples of pseudocompact groups $G_1$ $G_{1}$ and $G_2$ $G_{2}$ (in fact, they are even countably compact and of weight $\aleph _2$ $ℵ_{2}$ ) such that $G_1$ $G_{1}$ is Ascoli but not a $k_\mathbb {R}$ $k_{R}$ -space, and $G_2$ $G_{2}$ is a $k_\mathbb {R}$ $k_{R}$ -space but not a k-space. Under $\mathrm {MA+\lnot CH}$ $MA + \neg CH$ , we show that any pseudocompact group of weight $\aleph _1$ $ℵ_{1}$ is Ascoli. These results are proved using topological properties of pseudocompact spaces X of weight $\aleph _1$ $ℵ_{1}$ and of $\Sigma $ $Σ$ -products in products of compact spaces. Being motivated by these results and the countably compact part of Noble’s theorem, it is shown by a well-known technique that each countably compact infinite group has a separable countably compact subgroup of cardinality continuum.

Closed embeddings of spaces and groups into pseudocompact \(k_\mathbb {R}\)-groups

摘要