Definition:Sober Space

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Definition

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 $\set x$.


Separation Axiom vs Completeness Property

In some sources sobriety is listed among the separation axioms.

It has been pointed out in other sources that sobriety is a property of completeness type, similar to the completeness of metric spaces.

From Sober Space iff Completely Prime Filter is Unique System of Open Neighborhoods, filters of open sets that look like systems of open neighborhoods are required to really be systems of open neighborhoods.


Also see

  • Results about sober spaces can be found here.