On generic topological embeddings
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866914860213731328 |
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| author | Kubiś, Wiesław Kucharski, Andrzej Turek, Sławomir |
| author_facet | Kubiś, Wiesław Kucharski, Andrzej Turek, Sławomir |
| contents | We show that an embedding of a fixed 0-dimensional compact space $K$ into the Čech--Stone remainder $ω^*$ as a nowhere dense P-set is the unique generic limit, a special object in the category consisting of all continuous maps from $K$ to compact metric spaces.
Using Fraïssé theory we get a few well know theorems about Čech--Stone remainder.
We establish the following:
-- an ultrametric space $K$ of weight $κ$ can be uniformly embedded into $κ^κ$ as a uniformly nowhere dense subset,
-- every uniform homeomorphism of uniformly nowhere dense sets in $κ^κ$
can be extended to a uniform auto-homeomorphism of $κ^κ$,
-- every uniformly nowhere dense set in $κ^κ$ is a uniform retract of $κ^κ$.
If we assume that $κ$ is a weakly compact cardinal we get the counterpart of the above result without the uniformity assumption. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_05043 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On generic topological embeddings Kubiś, Wiesław Kucharski, Andrzej Turek, Sławomir General Topology 54B30, 18F60, 54C25, 54C15 We show that an embedding of a fixed 0-dimensional compact space $K$ into the Čech--Stone remainder $ω^*$ as a nowhere dense P-set is the unique generic limit, a special object in the category consisting of all continuous maps from $K$ to compact metric spaces. Using Fraïssé theory we get a few well know theorems about Čech--Stone remainder. We establish the following: -- an ultrametric space $K$ of weight $κ$ can be uniformly embedded into $κ^κ$ as a uniformly nowhere dense subset, -- every uniform homeomorphism of uniformly nowhere dense sets in $κ^κ$ can be extended to a uniform auto-homeomorphism of $κ^κ$, -- every uniformly nowhere dense set in $κ^κ$ is a uniform retract of $κ^κ$. If we assume that $κ$ is a weakly compact cardinal we get the counterpart of the above result without the uniformity assumption. |
| title | On generic topological embeddings |
| topic | General Topology 54B30, 18F60, 54C25, 54C15 |
| url | https://arxiv.org/abs/2310.05043 |