Definition:Sober Space/Definition 2

Definition

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

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

for every meet-irreducible open set $U \ne S$ there exists a unique $x \in S$ such that:

$U = S \setminus \set x^-$

where $\set x^-$ denotes the closure of $\set x$.

Also see

• Equivalence of Definitions of Sober Space

• Complement of Singleton Closure is Meet-Irreducible

• Results about sober spaces can be found here.

Sources

• 2012: Jorge Picado and Aleš Pultr: Frames and Locales: Chapter $1$: Spaces and Lattices of Open Sets, $\S 1$ Sober spaces, Definition $1.1$