Topological size of the set of universal and ultrahomogeneous retractions on the Urysohn space
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866908943850143744 |
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| author | Bąk, Judyta Garbulińska-Węgrzyn, Joanna Popławski, Michał |
| author_facet | Bąk, Judyta Garbulińska-Węgrzyn, Joanna Popławski, Michał |
| contents | In this paper, we investigate the set $\mathcal{U}(\mathbb{U})$ of universal and ultrahomogeneous $1$-Lipschitz retractions acting on the Urysohn space as the subspace of the space $\mathcal{R}(\mathbb{U})$ of all $1-$Lipschitz retractions defined on the Urysohn space. Especially, we study Borel complexity and density $\mathcal{U}(\mathbb{U})$ in $\mathcal{R}(\mathbb{U}).$ In order to do that, we introduce a new extension property $(UR^*)$ that is equivalent to the universality and ultrahomogeneity of a retraction, and a new pointwise retract topology. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_05813 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Topological size of the set of universal and ultrahomogeneous retractions on the Urysohn space Bąk, Judyta Garbulińska-Węgrzyn, Joanna Popławski, Michał General Topology 54E50, 18A30, 18A35, 54C35 In this paper, we investigate the set $\mathcal{U}(\mathbb{U})$ of universal and ultrahomogeneous $1$-Lipschitz retractions acting on the Urysohn space as the subspace of the space $\mathcal{R}(\mathbb{U})$ of all $1-$Lipschitz retractions defined on the Urysohn space. Especially, we study Borel complexity and density $\mathcal{U}(\mathbb{U})$ in $\mathcal{R}(\mathbb{U}).$ In order to do that, we introduce a new extension property $(UR^*)$ that is equivalent to the universality and ultrahomogeneity of a retraction, and a new pointwise retract topology. |
| title | Topological size of the set of universal and ultrahomogeneous retractions on the Urysohn space |
| topic | General Topology 54E50, 18A30, 18A35, 54C35 |
| url | https://arxiv.org/abs/2604.05813 |