Category:Sober Spaces

This category contains results about Sober Spaces.
Definitions specific to this category can be found in Definitions/Sober Spaces.

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

Definition 1

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

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

Definition 2

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 $\set x$.