The decomposable and the ambiguous sets

Abstract

We prove that every ambiguous subset of a hereditarily Baire space is decomposable. We obtain that a decomposable set $A\subseteq X$ is ambiguous when (i) $X$ is a perfectly paracompact space, or (ii) $A$ and $X\setminus A$ are Lindelöf and $X$ is a completely regular space.