We say that a Tychonoff space X is a <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\kappa $</EquationSource> <EquationSource Format="MATHML"><math> <mi>κ</mi> </math></EquationSource> </InlineEquation>-space if it is homeomorphic to a closed subspace of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$C_p(Y)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>C</mi> <mi>p</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>Y</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> for some locally compact space Y. The class of <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\kappa $</EquationSource> <EquationSource Format="MATHML"><math> <mi>κ</mi> </math></EquationSource> </InlineEquation>-spaces is strictly between the class of Dieudonné complete spaces and the class of <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\mu $</EquationSource> <EquationSource Format="MATHML"><math> <mi>μ</mi> </math></EquationSource> </InlineEquation>-spaces. We show that the class of <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\kappa $</EquationSource> <EquationSource Format="MATHML"><math> <mi>κ</mi> </math></EquationSource> </InlineEquation>-spaces has nice stability properties, that allows us to define the <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\kappa $</EquationSource> <EquationSource Format="MATHML"><math> <mi>κ</mi> </math></EquationSource> </InlineEquation>-completion <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\kappa X$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>κ</mi> <mi>X</mi> </mrow> </math></EquationSource> </InlineEquation> of X as the smallest <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$\kappa $</EquationSource> <EquationSource Format="MATHML"><math> <mi>κ</mi> </math></EquationSource> </InlineEquation>-space in the Stone–Čech compactification <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$\beta X$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>β</mi> <mi>X</mi> </mrow> </math></EquationSource> </InlineEquation> of X containing X. For a point <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$z\in \beta X$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>z</mi> <mo>∈</mo> <mi>β</mi> <mi>X</mi> </mrow> </math></EquationSource> </InlineEquation>, we show that (1) if <InlineEquation ID="IEq14"> <EquationSource Format="TEX">$z\in \upsilon X$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>z</mi> <mo>∈</mo> <mi>υ</mi> <mi>X</mi> </mrow> </math></EquationSource> </InlineEquation>, then the evaluation function <InlineEquation ID="IEq15"> <EquationSource Format="TEX">$\delta _z$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>δ</mi> <mi>z</mi> </msub> </math></EquationSource> </InlineEquation> at z is bounded on each compact subset of <InlineEquation ID="IEq16"> <EquationSource Format="TEX">$C_p(X)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>C</mi> <mi>p</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, (2) <InlineEquation ID="IEq17"> <EquationSource Format="TEX">$z\in \kappa X$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>z</mi> <mo>∈</mo> <mi>κ</mi> <mi>X</mi> </mrow> </math></EquationSource> </InlineEquation> iff <InlineEquation ID="IEq18"> <EquationSource Format="TEX">$\delta _z$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>δ</mi> <mi>z</mi> </msub> </math></EquationSource> </InlineEquation> is continuous on each compact subset of <InlineEquation ID="IEq19"> <EquationSource Format="TEX">$C_p(X)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>C</mi> <mi>p</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> iff <InlineEquation ID="IEq20"> <EquationSource Format="TEX">$\delta _z$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>δ</mi> <mi>z</mi> </msub> </math></EquationSource> </InlineEquation> is continuous on each compact subset of <InlineEquation ID="IEq21"> <EquationSource Format="TEX">$C_p^b(X)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>C</mi> <mi>p</mi> <mi>b</mi> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, (3) <InlineEquation ID="IEq22"> <EquationSource Format="TEX">$z\in \upsilon X$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>z</mi> <mo>∈</mo> <mi>υ</mi> <mi>X</mi> </mrow> </math></EquationSource> </InlineEquation> iff <InlineEquation ID="IEq23"> <EquationSource Format="TEX">$\delta _z$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>δ</mi> <mi>z</mi> </msub> </math></EquationSource> </InlineEquation> is bounded on each compact subset of <InlineEquation ID="IEq24"> <EquationSource Format="TEX">$C_p^b(X)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>C</mi> <mi>p</mi> <mi>b</mi> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. It is proved that <InlineEquation ID="IEq25"> <EquationSource Format="TEX">$\kappa X$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>κ</mi> <mi>X</mi> </mrow> </math></EquationSource> </InlineEquation> is the largest subspace Y of <InlineEquation ID="IEq26"> <EquationSource Format="TEX">$\beta X$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>β</mi> <mi>X</mi> </mrow> </math></EquationSource> </InlineEquation> containing X for which <InlineEquation ID="IEq27"> <EquationSource Format="TEX">$C_p(Y)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>C</mi> <mi>p</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>Y</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq28"> <EquationSource Format="TEX">$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> have the same compact subsets, this result essentially generalizes a known result of R. Haydon.

$\kappa $-spaces

期刊论文
发表日期：

2026年3月9日

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Saak Gabriyelyan,
Evgenii Reznichenko

摘要

We say that a Tychonoff space X is a $\kappa $ $κ$ -space if it is homeomorphic to a closed subspace of $C_p(Y)$ $C_{p} (Y)$ for some locally compact space Y. The class of $\kappa $ $κ$ -spaces is strictly between the class of Dieudonné complete spaces and the class of $\mu $ $μ$ -spaces. We show that the class of $\kappa $ $κ$ -spaces has nice stability properties, that allows us to define the $\kappa $ $κ$ -completion $\kappa X$ $κ X$ of X as the smallest $\kappa $ $κ$ -space in the Stone–Čech compactification $\beta X$ $β X$ of X containing X. For a point $z\in \beta X$ $z \in β X$ , we show that (1) if $z\in \upsilon X$ $z \in υ X$ , then the evaluation function $\delta _z$ $δ_{z}$ at z is bounded on each compact subset of $C_p(X)$ $C_{p} (X)$ , (2) $z\in \kappa X$ $z \in κ X$ iff $\delta _z$ $δ_{z}$ is continuous on each compact subset of $C_p(X)$ $C_{p} (X)$ iff $\delta _z$ $δ_{z}$ is continuous on each compact subset of $C_p^b(X)$ $C_{p}^{b} (X)$ , (3) $z\in \upsilon X$ $z \in υ X$ iff $\delta _z$ $δ_{z}$ is bounded on each compact subset of $C_p^b(X)$ $C_{p}^{b} (X)$ . It is proved that $\kappa X$ $κ X$ is the largest subspace Y of $\beta X$ $β X$ containing X for which $C_p(Y)$ $C_{p} (Y)$ and $C_p(X)$ $C_{p} (X)$ have the same compact subsets, this result essentially generalizes a known result of R. Haydon.

\(\kappa \)-spaces

摘要