Irreducible Locales

Abstract

Basic properties of irreducible locales which extend results contained in ~\cite{Du} are presented. Our main result is that every locale $L$ can be embedded as a closed nowhere dense sublocale of an irreducible locale $\mathcal{I}L$, what we call the \textit{irreducible envelope} of $L$. The properties
of \textit{spatiality, subfitness, fitness, compactness,} and the \textit{Noetherian} property are shown to be inherited and reflected by the irreducible envelope.