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| Main Author: | |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2402.11213 |
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| _version_ | 1866913961275817984 |
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| author | Lipin, Anton |
| author_facet | Lipin, Anton |
| contents | We investigate connections between resolvability and different forms of tightness. This study is adjacent to [1,2].
We construct a non-regular refinement $τ^*$ of the natural topology of the real line $\mathbb{R}$ with properties such that the space $(\mathbb{R}, τ^*)$ has a hereditary nowhere dense tightness and it has no $ω_1$-resolvable subspaces, whereas $Δ(\mathbb{R}, τ^*) = \frak{c}$.
We also show that the proof of the main result of [1], being slightly modified, leads to the following strengthening: if $L$ is a Hausdorff space of countable character and the space $L^ω$ is c.c.c., then every submaximal dense subspace of $L^κ$ has disjoint tightness. As a corollary, for every $κ\geq ω$ there is a Tychonoff submaximal space $X$ such that $|X|=Δ(X)=κ$ and $X$ has disjoint tightness. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_11213 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On resolvability and tightness in uncountable spaces Lipin, Anton General Topology We investigate connections between resolvability and different forms of tightness. This study is adjacent to [1,2]. We construct a non-regular refinement $τ^*$ of the natural topology of the real line $\mathbb{R}$ with properties such that the space $(\mathbb{R}, τ^*)$ has a hereditary nowhere dense tightness and it has no $ω_1$-resolvable subspaces, whereas $Δ(\mathbb{R}, τ^*) = \frak{c}$. We also show that the proof of the main result of [1], being slightly modified, leads to the following strengthening: if $L$ is a Hausdorff space of countable character and the space $L^ω$ is c.c.c., then every submaximal dense subspace of $L^κ$ has disjoint tightness. As a corollary, for every $κ\geq ω$ there is a Tychonoff submaximal space $X$ such that $|X|=Δ(X)=κ$ and $X$ has disjoint tightness. |
| title | On resolvability and tightness in uncountable spaces |
| topic | General Topology |
| url | https://arxiv.org/abs/2402.11213 |