nLab spatial sigma-locale

Idea

The analogue of a spatial locale for $\sigma$-locales

Definitions

Let $X$ be a $\sigma$-topological space. Then we may define a $\sigma$-locale, denoted $O(X)$, whose $\sigma$-frame of opens is precisely the $\sigma$-frame of open subspaces of $X$.

A locale is spatial or topological if it is isomorphic to $O(X)$ for some $\sigma$-topological space $X$.

A locale $L$ has enough points if, given any two opens $U$ and $V$ in $L$, $U = V$ if (hence iff) precisely the same points of $L$ belong to $U$ as belong to $V$.

Related concepts

• sigma-locale

• spatial locale