  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).
