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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2505.18704 |
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| _version_ | 1866918084637360128 |
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| author | Lipin, Anton |
| author_facet | Lipin, Anton |
| contents | Suppose $X$ and $Y$ are topological spaces, $|X| = Δ(X)$ and $|Y| = Δ(Y)$. We investigate resolvability of the product $X \times Y$. We prove that: I. If $|X| = |Y| = ω$ and $X,Y$ are Hausdorff, then $X \times Y$ is maximally resolvable; II. If $2^κ= κ^+$, $\{|X|, \mathrm{cf}|X|\} \cap \{κ, κ^+\} \ne \emptyset$ and $\mathrm{cf}|Y| = κ^+$, then the space $X \times Y$ is $κ^+$-resolvable. In particular, under GCH the space $X^2$ is $\mathrm{cf}|X|$-resolvable whenever $\mathrm{cf}|X|$ is an isolated cardinal; III. ($\frak{r} = \frak{c}$) If $\mathrm{cf}|X| = ω$ and $\mathrm{cf}|Y| = \mathrm{cf}(\frak{c})$, then the space $X \times Y$ is $ω$-resolvable. If, moreover, $\mathrm{cf}(\frak{c}) = ω_1$, then the space $X \times Y$ is $ω_1$-resolvable. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_18704 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Resolvability in products and squares Lipin, Anton General Topology Suppose $X$ and $Y$ are topological spaces, $|X| = Δ(X)$ and $|Y| = Δ(Y)$. We investigate resolvability of the product $X \times Y$. We prove that: I. If $|X| = |Y| = ω$ and $X,Y$ are Hausdorff, then $X \times Y$ is maximally resolvable; II. If $2^κ= κ^+$, $\{|X|, \mathrm{cf}|X|\} \cap \{κ, κ^+\} \ne \emptyset$ and $\mathrm{cf}|Y| = κ^+$, then the space $X \times Y$ is $κ^+$-resolvable. In particular, under GCH the space $X^2$ is $\mathrm{cf}|X|$-resolvable whenever $\mathrm{cf}|X|$ is an isolated cardinal; III. ($\frak{r} = \frak{c}$) If $\mathrm{cf}|X| = ω$ and $\mathrm{cf}|Y| = \mathrm{cf}(\frak{c})$, then the space $X \times Y$ is $ω$-resolvable. If, moreover, $\mathrm{cf}(\frak{c}) = ω_1$, then the space $X \times Y$ is $ω_1$-resolvable. |
| title | Resolvability in products and squares |
| topic | General Topology |
| url | https://arxiv.org/abs/2505.18704 |