Suppose <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$X$</EquationSource> <EquationSource Format="MATHML"><math> <mi>X</mi> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$Y$</EquationSource> <EquationSource Format="MATHML"><math> <mi>Y</mi> </math></EquationSource> </InlineEquation> are topological spaces, <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$|X| = \Delta(X)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">|</mo> <mi>X</mi> <mo stretchy="false">|</mo> <mo>=</mo> <mi mathvariant="normal">Δ</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$|Y| = \Delta(Y)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">|</mo> <mi>Y</mi> <mo stretchy="false">|</mo> <mo>=</mo> <mi mathvariant="normal">Δ</mi> <mo stretchy="false">(</mo> <mi>Y</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. We investigate resolvability of the product <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$X \times Y$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>X</mi> <mo>×</mo> <mi>Y</mi> </mrow> </math></EquationSource> </InlineEquation>. We prove that: <UnorderedList Mark="None"> <ItemContent> I. If <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$|X| = |Y| = \omega$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">|</mo> <mi>X</mi> <mo stretchy="false">|</mo> <mo>=</mo> <mo stretchy="false">|</mo> <mi>Y</mi> <mo stretchy="false">|</mo> <mo>=</mo> <mi>ω</mi> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$X,Y$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>X</mi> <mo>,</mo> <mi>Y</mi> </mrow> </math></EquationSource> </InlineEquation> are Hausdorff, then <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$ { X \times Y } $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>X</mi> <mo>×</mo> <mi>Y</mi> </mrow> </math></EquationSource> </InlineEquation> is maximally resolvable. </ItemContent> <ItemContent> II. If <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$2^\kappa = \kappa^+$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mn>2</mn> <mi>κ</mi> </msup> <mo>=</mo> <msup> <mi>κ</mi> <mo>+</mo> </msup> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\{|X|, \mathrm{cf}|X|\} \cap \{\kappa, \kappa^+\} \ne \emptyset$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mo stretchy="false">{</mo> <mo stretchy="false">|</mo> <mi>X</mi> <mo stretchy="false">|</mo> <mo>,</mo> <mi mathvariant="normal">cf</mi> <mo stretchy="false">|</mo> <mi>X</mi> <mo stretchy="false">|</mo> <mo stretchy="false">}</mo> </mrow> <mo>∩</mo> <mrow> <mo stretchy="false">{</mo> <mi>κ</mi> <mo>,</mo> <msup> <mi>κ</mi> <mo>+</mo> </msup> <mo stretchy="false">}</mo> </mrow> <mo>≠</mo> <mi mathvariant="normal">∅</mi> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$\mathrm{cf}|Y| = \kappa^+$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mi mathvariant="normal">cf</mi> <mo stretchy="false">|</mo> <mi>Y</mi> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <msup> <mi>κ</mi> <mo>+</mo> </msup> </mrow> </math></EquationSource> </InlineEquation>, then the space <InlineEquation ID="IEq12"> <EquationSource Format="TEX">${X \times Y}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>X</mi> <mo>×</mo> <mi>Y</mi> </mrow> </math></EquationSource> </InlineEquation> is <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$\kappa^+$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>κ</mi> <mo>+</mo> </msup> </math></EquationSource> </InlineEquation>-resolvable. In particular, under GCH the space <InlineEquation ID="IEq14"> <EquationSource Format="TEX">$X^2$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>X</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation> is <InlineEquation ID="IEq15"> <EquationSource Format="TEX">$\mathrm{cf}|X|$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">cf</mi> <mo stretchy="false">|</mo> <mi>X</mi> <mo stretchy="false">|</mo> </mrow> </math></EquationSource> </InlineEquation>-resolvable whenever <InlineEquation ID="IEq16"> <EquationSource Format="TEX">$\mathrm{cf}|X|$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">cf</mi> <mo stretchy="false">|</mo> <mi>X</mi> <mo stretchy="false">|</mo> </mrow> </math></EquationSource> </InlineEquation> is an isolated cardinal. </ItemContent> <ItemContent> III. (<InlineEquation ID="IEq91"> <EquationSource Format="TEX">$\mathfrak{r} = \mathfrak{c}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="fraktur">r</mi> <mo>=</mo> <mi mathvariant="fraktur">c</mi> </mrow> </math></EquationSource> </InlineEquation>) If <InlineEquation ID="IEq17"> <EquationSource Format="TEX">$\mathrm{cf}|X| = \omega$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">cf</mi> <mo stretchy="false">|</mo> <mi>X</mi> <mo stretchy="false">|</mo> <mo>=</mo> <mi>ω</mi> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq18"> <EquationSource Format="TEX">${\rm cf}|Y| = {\rm cf}(\mathfrak{c})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">cf</mi> <mo stretchy="false">|</mo> <mi>Y</mi> <mo stretchy="false">|</mo> <mo>=</mo> <mi mathvariant="normal">cf</mi> <mo stretchy="false">(</mo> <mi mathvariant="fraktur">c</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, then the space <InlineEquation ID="IEq19"> <EquationSource Format="TEX">$ { X \times Y } $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>X</mi> <mo>×</mo> <mi>Y</mi> </mrow> </math></EquationSource> </InlineEquation> is <InlineEquation ID="IEq20"> <EquationSource Format="TEX">$\omega$</EquationSource> <EquationSource Format="MATHML"><math> <mi>ω</mi> </math></EquationSource> </InlineEquation>-resolvable. If, moreover, <InlineEquation ID="IEq21"> <EquationSource Format="TEX">${\rm cf}(\mathfrak{c}) = \omega_1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">cf</mi> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="fraktur">c</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msub> <mi>ω</mi> <mn>1</mn> </msub> </mrow> </math></EquationSource> </InlineEquation>, then the space <InlineEquation ID="IEq22"> <EquationSource Format="TEX">$ { X \times Y } $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>X</mi> <mo>×</mo> <mi>Y</mi> </mrow> </math></EquationSource> </InlineEquation> is <InlineEquation ID="IEq23"> <EquationSource Format="TEX">$\omega_1$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>ω</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation>-resolvable. </ItemContent> </UnorderedList>

Resolvability in products and squares

期刊论文
发表日期：

2025年10月16日

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A. E. Lipin

摘要

Suppose $X$ $X$ and $Y$ $Y$ are topological spaces, $|X| = \Delta(X)$ $| X | = Δ (X)$ and $|Y| = \Delta(Y)$ $| Y | = Δ (Y)$ . We investigate resolvability of the product $X \times Y$ $X \times Y$ . We prove that:

I. If $|X| = |Y| = \omega$ $| X | = | Y | = ω$ and $X,Y$ $X, Y$ are Hausdorff, then $ { X \times Y } $ $X \times Y$ is maximally resolvable.

II. If $2^\kappa = \kappa^+$ $2^{κ} = κ^{+}$ , $\{|X|, \mathrm{cf}|X|\} \cap \{\kappa, \kappa^+\} \ne \emptyset$ ${| X |, cf | X |} \cap {κ, κ^{+}} \neq \emptyset$ and $\mathrm{cf}|Y| = \kappa^+$ $cf | Y | = κ^{+}$ , then the space ${X \times Y}$ $X \times Y$ is $\kappa^+$ $κ^{+}$ -resolvable. In particular, under GCH the space $X^2$ $X^{2}$ is $\mathrm{cf}|X|$ $cf | X |$ -resolvable whenever $\mathrm{cf}|X|$ $cf | X |$ is an isolated cardinal.

III. ( $\mathfrak{r} = \mathfrak{c}$ $r = c$ ) If $\mathrm{cf}|X| = \omega$ $cf | X | = ω$ and ${\rm cf}|Y| = {\rm cf}(\mathfrak{c})$ $cf | Y | = cf (c)$ , then the space $ { X \times Y } $ $X \times Y$ is $\omega$ $ω$ -resolvable. If, moreover, ${\rm cf}(\mathfrak{c}) = \omega_1$ $cf (c) = ω_{1}$ , then the space $ { X \times Y } $ $X \times Y$ is $\omega_1$ $ω_{1}$ -resolvable.