A generalization of Whyburn's theorem, and aperiodicity for Abelian C⁎-inclusions

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