On Closed Mappings of Sigma-Compact Spaces and Dimension

2017/06/14 by Elżbieta Pol, Pol, Elżbieta, Roman Pol +1
Mathematics · #54D40 #54E40 #54F45 #54H05 #57N20 #Advanced Banach Space Theory #Advanced Topology and Set Theory #FOS: Mathematics #General Topology (math.GN) #Rings, Modules, and Algebras

paper · pdf · doi:10.48550/arxiv.1706.04398

openalex publication_date 2017/06/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove that if K is a remainder of the Hilbert space (i.e., K is the complement of the Hilbert space in its metrizable compactification) then every non-one-point closed image of K either contains a compact set with no transfinite dimension or contains compact sets of arbitrarily high inductive transfinite dimension ind. We construct also for each natural n a sigma-compact metrizable n-dimensional space whose image under any non-constant closed map has dimension at least n, and analogous examples for the transfinite dimension ind.

Citations

• Isometric embeddings and continuous maps onto the irrationals

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