On generic topological embeddings

2023/10/08 by Wiesław Kubiś, Andrzej Kucharski, Kubiś, Wiesław +3
Mathematics · #18F60 #54B30 #54C15 #54C25 #Advanced Topology and Set Theory #FOS: Mathematics #General Topology (math.GN) #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.2310.05043

openalex publication_date 2023/10/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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.

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