A note on uniform continuity of monotone functions
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866915176656142336 |
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| author | Pol, Roman Zakrzewski, Piotr Zdomskyy, Lyubomyr |
| author_facet | Pol, Roman Zakrzewski, Piotr Zdomskyy, Lyubomyr |
| contents | We prove that it is consistent with ZFC that for every non-decreasing function $f:[0,1]\to [0,1]$, each subset of $[0,1]$ of cardinality $\mathfrak c$ contains a set of cardinality $\mathfrak c$ on which $f$ is uniformly continuous. We show that this statement follows from the assumptions that $\mathfrak d^* < \mathfrak c$ and $\mathfrak c$ is regular, where $\mathfrak d^*\leq \mathfrak d$ is the smallest cardinality $κ$ such that any two disjoint countable dense sets in the Cantor set can be separated by sets each of which is an intersection of at most $κ$-many open sets in the Cantor set. We establish also that $\mathfrak d^*=\min\{\mathfrak u, \mathfrak d\}=\min\{\mathfrak r, \mathfrak d\}$, thus giving an alternative proof of the latter equality established by J. Aubrey in 2004. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_20887 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A note on uniform continuity of monotone functions Pol, Roman Zakrzewski, Piotr Zdomskyy, Lyubomyr Logic General Topology 03E20, 03E17, 26A15, 54C05, 54E45 We prove that it is consistent with ZFC that for every non-decreasing function $f:[0,1]\to [0,1]$, each subset of $[0,1]$ of cardinality $\mathfrak c$ contains a set of cardinality $\mathfrak c$ on which $f$ is uniformly continuous. We show that this statement follows from the assumptions that $\mathfrak d^* < \mathfrak c$ and $\mathfrak c$ is regular, where $\mathfrak d^*\leq \mathfrak d$ is the smallest cardinality $κ$ such that any two disjoint countable dense sets in the Cantor set can be separated by sets each of which is an intersection of at most $κ$-many open sets in the Cantor set. We establish also that $\mathfrak d^*=\min\{\mathfrak u, \mathfrak d\}=\min\{\mathfrak r, \mathfrak d\}$, thus giving an alternative proof of the latter equality established by J. Aubrey in 2004. |
| title | A note on uniform continuity of monotone functions |
| topic | Logic General Topology 03E20, 03E17, 26A15, 54C05, 54E45 |
| url | https://arxiv.org/abs/2502.20887 |