A Generalization of Whyburn's Theorem, and Aperiodicity for Abelian C*-Inclusions

Let $j:Y \to X$ be a continuous surjection of compact metric spaces. Whyburn proved that $j$ is irreducible, meaning that $j(F) \subsetneq X$ for any proper closed subset $F \subsetneq Y$, if and only if $j$ is almost one-to-one, in the sense that $$\overline{\{y \in Y: j^{-1}(j(y)) = y\}} = Y.$$ In this note we prove the following generalization: There exists a unique minimal closed set $K \subseteq Y$ such that $j(K) = X$ if and only if $$\overline{\{x \in X: card(j^{-1}(x)) = 1\}} = X.$$ Translated to the language of operator algebras, this says that if $A \subseteq B$ is a unital inclusion of separable abelian $C^*$-algebras, then there exists a unique pseudo-expectation (in the sense of Pitts) if and only if the almost extension property of Nagy-Reznikoff holds. More generally, we prove that a unital inclusion of (not necessarily separable) abelian $C^*$-algebras has a unique pseudo-expectation if and only if it is aperiodic (in the sense of Kwa\'{s}niewski-Meyer).