Eventually Constant and stagnating functions in non-Lindel\"of spaces

Abstract

Inspired by recent work of A. Mardani which elaborates on the elementary fact that for any continuous function $f:\omega_1\times\mathbb{R}\to\mathbb{R}$, there is an $\alpha\in\omega_1$ such that $f(\langle\beta,x\rangle) = f(\langle\alpha,x\rangle)$ for all $\beta\ge\alpha$ and $x\in\mathbb{R}$, we introduce four properties $\mathsf{P}(X,Y)$, $\mathsf{P}\in\{\mathsf{EC},\mathsf{S},\mathsf{L},\mathsf{BR}\}$, which are different formalizations of 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". We say that the spaces $X,Y$ satisfy the property $\mathsf{EC}(X,Y)$ (resp. $\mathsf{S}(X,Y)$) [resp. $\mathsf{L}(X,Y)$] iff given $f:X\to Y$, then there is a Lindel\"of $Z\subset X$ 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.) We investigate the relations between these four and other classical topological properties. Two variants of each property are given depending on whether $Z$ can be chosen to be closed. Here is a sample of our results. An uncountable subspace $T$ of a tree of height $\omega_1$ is $\omega_1$-compact iff $\mathsf{S}(T,Y)$ holds for any metrizable space $Y$ of cardinality $>1$. 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 $\omega_1$-compact. The property $\mathsf{L}(M,\mathbb{R})$ holds for any manifold while $\mathsf{L}(M,\mathbb{R}^2)$ does not. Under PFA, a locally compact countably tight space $Y$ for which $\mathsf{EC}(\omega_1,Y)$ holds is isocompact, while there are counterexamples under $\clubsuit_C$. Some of our results are restatements of other researchers work put in our context.