Eventually Constant and stagnating functions in non-Lindelöf spaces

Abstract

We elaborate on the elementary fact that for any continuous function $f:ω_1\times\mathbb{R}\to\mathbb{R}$, there is an $α\inω_1$ such that $f(\langleβ,x\rangle) = f(\langleα,x\rangle)$ for all $β\geα$ 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 $\mathsf{P_{cl}},\mathsf{P_{cpt}}$ 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 $ω_1$ is $ω_1$-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 $ω_1$-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 $\mathsf{EC}(ω_1,Y)$ holds is isocompact, while there are counterexamples under $\clubsuit_C$. Some of our results are (more or less elaborate) restatements of other researchers work put in our context.