Equivalence of Definitions of Sober Space

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Theorem

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

The following definitions of the concept of Sober Space are equivalent:

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 $\set x$.

Proof

Definition 1 implies Definition 2

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

Let $U \ne S$ be a meet-irreducible open set.

Let $F = S \setminus U$.

From Meet-Irreducible Open Set iff Complement is Closed Irreducible Subspace:

$F$ is closed irreducible subspace

We have by hypothesis:

$\exists ! x \in S : F = \set x^-$

From Relative Complement inverts Subsets of Relative Complement:

$\exists ! x \in S : U = S \setminus \set x^-$

The result follows.

$\Box$

Definition 2 implies Definition 1

Let for every meet-irreducible open set $V \ne S$ there exists a unique $y \in S$ such that:

$V = S \setminus \set y^-$

Let $F$ closed irreducible subspace of $T$.

Let $U = S \setminus F$.

From Meet-Irreducible Open Set iff Complement is Closed Irreducible Subspace:

$U$ is a meet-irreducible open set

We have by hypothesis:

$\exists ! x \in S : U = S \setminus \set x^-$

From Relative Complement inverts Subsets of Relative Complement:

$\exists ! x \in S : F = \set x^-$

The result follows.

$\blacksquare$