nLab sober sigma-topological space

Idea

The analogue of a sober topological space for $\sigma$-topological spaces.

Definition

A $\sigma$-topological space $X$ is sober if its points are exactly determined by its lattice of open subsets. Different equivalent ways to say this are:

• The continuous map from $X$ to the space of points of the $\sigma$-locale that it gives rise to (see there for details) is a homeomorphism.

• The function from points of $X$ to the countably prime filters of its open-set lattice is a bijection.

Related concepts

• sober topological space

• sigma-topological space

• sigma-locale