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 ⊂ X Y \subset X is connected and nondegenerate iff X ∖ Y 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.