2023/08/24 by Mathieu Baillif, Baillif, Mathieu
Decision Sciences · Mathematics · #Advanced Topology and Set Theory #Fuzzy and Soft Set Theory #math.GN
paper · pdf · doi:10.48550/arxiv.2308.12763
openalex publication_date 2023/08/24 · openalex created_date 2023/08/26 · openalex updated_date 2026/07/28
We elaborate on the elementary fact that for any continuous function f:ω1×ℝ→ℝ, there is an α∈ω1 such that f(⟨β,x⟩) = f(⟨α,x⟩) for all β≥α and x∈ℝ, we introduce four properties P(X,Y), P∈\EC,S,L,BR\ generalizating Lindelöfness, which formalize the idea vaguely stated as ``given a continuous f:X→ Y, there is a small subspace of X outside of which f does not do anything much new''. The spaces X,Y satisfy EC(X,Y) [resp. S(X,Y)] (resp. L(X,Y)) iff given f:X→ Y, then there is a Lindelöf Z⊂ X such that f(X-Z) is a singleton [resp. there is a retraction r:X→ Z such that f∘ r = f] (resp. f(Z) = f(X)). BR(X,Y) is defined similarly. Two more variants \mathsfPcl,\mathsfPcpt of each property are given depending on whether Z can be chosen to be closed or compact. We investigate the relations between these and other classical topological properties. Here is a sample of our results. An uncountable subspace T of a tree of height ω1 is ω1-compact iff S(T,Y) holds for any metrizable space Y of uncountable cardinality. If M is a ℵ1-strongly collectionwise Hausforff non-metrizable manifold satisfying either (a weakening of) S(M,ℝ) or EC(M,ℝ), then M is ω1-compact. L(M,ℝ) holds for any manifold while L(M,ℝ2) does not. Under \bf PFA, a locally compact countably tight space Y for which EC(ω1,Y) holds is isocompact, while there are counterexamples under \clubsuitC. Some of our results are (more or less elaborate) restatements of other researchers work put in our context.