Spaces in which the nondegenerate connected sets are the cofinite sets

Abstract:

Assuming the continuum hypothesis (CH), we construct a perfectly normal space X such that $Y \subset X$ is connected and nondegenerate iff $X\backslash Y$ is finite. We also show that completely regular, as well as countable Hausdorff, examples of this kind can be obtained under axioms weaker than CH, e.g., Martin’s Axiom.