Construction of a Subspace of the Spectrum of L From the L-Slice for a Locale L

Abstract

The notion of compactness in the L-slice $(\sigma,J)$ of a locale $ L $ is introduced and we show that L-slice compactness is stronger than topological compactness and localic compactness. A subspace $ Y $, of the spectrum $Sp(L)$ of the locale $ L $, is constructed using filters $F_{x}=\{a \in L: \sigma(a,x)=x\}$ for compact elements $ x \in(\sigma,J)$ and the compactness of the subspace $ Y $ is characterised using the existence of a maximal compact element in the L-slice $ (\sigma,J) $.