Eventually Constant and stagnating functions in non-Lindelöf spaces

Abstract

We elaborate on the elementary fact that for any continuous function $f:\ddot{I}{‰}_1\times \mathbb{R}\to \mathbb{R}$, there is an $\hat{I}\pm \in \ddot{I}{‰}_1$ such that $f(⟨{\hat{I}}^2,x⟩)=f(⟨\hat{I}\pm ,x⟩)$ for all ${\hat{I}}^2\ge \hat{I}\pm$ and $x\in \mathbb{R}$, we introduce four properties $\mathsf{P}(X,Y)$, $\mathsf{P}\in \{\mathsf{EC},\mathsf{S},\mathsf{L},\mathsf{BR}\}$ generalizating Lindelöfness, which formalize the idea vaguely stated as ``given a continuous $f:X\to Y$, there is a small subspace of $X$ outside of which $f$ does not do anything much new''. The spaces $X,Y$ satisfy $\mathsf{EC}(X,Y)$ [resp. $\mathsf{S}(X,Y)$] (resp. $\mathsf{L}(X,Y)$) iff given $f:X\to Y$, then there is a Lindelöf $Z\subset X$ such that $f(X-Z)$ is a singleton [resp. there is a retraction $r:X\to Z$ such that $f\circ r=f$] (resp. $f(Z)=f(X)$). $\mathsf{BR}(X,Y)$ is defined similarly. Two more variants ${\mathsf{P}}_{\mathsf{cl}},{\mathsf{P}}_{\mathsf{cpt}}$ 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 $\ddot{I}{‰}_1$ is $\ddot{I}{‰}_1$-compact iff $\mathsf{S}(T,Y)$ holds for any metrizable space $Y$ of uncountable cardinality. If $M$ is a ${\aleph }_1$-strongly collectionwise Hausforff non-metrizable manifold satisfying either (a weakening of) $\mathsf{S}(M,\mathbb{R})$ or $\mathsf{EC}(M,\mathbb{R})$, then $M$ is $\ddot{I}{‰}_1$-compact. $\mathsf{L}(M,\mathbb{R})$ holds for any manifold while $\mathsf{L}(M,{\mathbb{R}}^2)$ does not. Under {\bf PFA}, a locally compact countably tight space $Y$ for which $\mathsf{EC}(\ddot{I}{‰}_1,Y)$ holds is isocompact, while there are counterexamples under ${♣}_C$. Some of our results are (more or less elaborate) restatements of other researchers work put in our context.

38 pages, 4 figures. V4: Mainly cosmetic corrections, added a last section containing questions