Definition:Sober Space/Definition 1

Definition

Let $T = \struct{S, \tau}$ be a topological space.

Then $T$ is a sober space if and only if:

each closed irreducible subspace of $T$ has a unique generic point.

Also see

• Equivalence of Definitions of Sober Space

• Results about sober spaces can be found here.

Sources

• 1982: Peter T. Johnstone: Stone Spaces: Chapter $\text {II}$: Introduction to locales, $\S 1.6$