Integer-valued functions and increasing unions of first countable spaces
Let \(\kappa\) be an infinite cardinal number, with \(\{X_{\alpha}:\alpha <\kappa \}\) a \(\kappa\)-chain of first countable spaces (i.e., \(X_{\alpha}\subseteq X_{\beta}\) for \(\alpha <\beta <\kappa)\). Let \(X=\cup_{\alpha <\kappa}X_{\alpha}\). Then there is a coarsest topology and a finest topology on X such that each \(X_{\alpha}\) is a subspace. Called the weak and fine topologies respectively in this paper, the question is raised as to when there is some compatible first countable topology on X. The following summarizes several interesting results the authors have obtained investigating this question. 1. If \(\kappa =\aleph_ 0\), then the weak topology is first countable and the fine topology is ``almost never'' first countable. 2. If \(\kappa\) is uncountable regular and X has a compatible first countable topology, then that topology must be the fine topology. 3. As for the main question, the one that asks for each uncountable regular \(\kappa\) whether the fine topology is first countable, the answer is absolutely ``no'' for \(\kappa =\aleph_ 1\), but is otherwise axiom- sensitive: The answer is ``yes'' if \(\kappa\) is either weakly compact or \(\geq\) some supercompact cardinal; if \(V=L\), then the answer is ``no'' unless \(\kappa\) is weakly compact. On the other hand, if there exists a supercompact cardinal, then forcing models can be constructed in which the answer is ``yes'' for every \(\kappa >\aleph_ 1.\) Several of the proofs exploit a translation of the main question into a mathematically equivalent question involving certain families of integer- valued functions.
scientific article; zbMATH DE number 4169625