On cardinalities and compact closures

Abstract

We show that there exists a Hausdorff topology on the set R of real numbers such that a subset A of R has compact closure if and only if A is countable. More generally, given any set X and any infinite set S, we prove that there exists a Hausdorff topology on X such that a subset A of X has compact closure if and only if the cardinality of A is less than or equal to that of S. When we attempt to replace “than than or equal to” in the preceding statement with “strictly less than,” the situation is more delicate; we show that the theorem extends to this case when S has regular cardinality but can fail when it does not. This counterexample shows that not every bornology is a bornology of compact closure. These results lie in the intersection of analysis, general topology, and set theory.