T2 Space is Sober Space

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Theorem

Let $\struct {S, \tau}$ be a $T_2$ (Hausdorff) space.


Then:

$\struct {S, \tau}$ is a sober space


Proof

Let $W \in \tau$ be meet-irreducible open proper subset of $S$.


Let $x \in S \setminus W$.

We have $S \setminus W$ is a closed set by definition.

From Closure of Subset of Closed Set of Topological Space is Subset:

$\set x^- \subseteq S \setminus W$

From Relative Complement inverts Subsets of Relative Complement:

$W \subseteq S \setminus \set x^-$

Hence:

$x \notin W$


Let $y \in S \setminus \set x^-$.

Then:

$y \ne x$


By definition of $T_2$ (Hausdorff) space:

$\exists W_1, W_2 \in \tau : x \in W_1, y \in W_2 : W_1 \cap W_2 = \O$

Let:

$U = W \cup W_1$

and

$V = W \cup W_2$

Hence:

$x \in U, y \in V$


Since $x \notin W$ and $x \in U$:

$U \nsubseteq W$


We have:

\(\ds U \cap V\) \(=\) \(\ds \paren {W \cup W_1} \cap \paren {W \cup W_2}\) Definition of $U$ and $V$
\(\ds \) \(=\) \(\ds \paren {W \cap W} \cup \paren {W \cap W_2} \cup \paren {W_1 \cap W} \cup \paren {W_1 \cap W_2}\) Intersection Distributes over Union
\(\ds \) \(=\) \(\ds W \cup \paren {W \cap W_2} \cup \paren {W_1 \cap W} \cup \paren {W_1 \cap W_2}\) Set Intersection is Idempotent
\(\ds \) \(=\) \(\ds W \cup \paren {W \cap W_2} \cup \paren {W_1 \cap W} \cup \O\) as $W_1 \cap W_2 = \O$
\(\ds \) \(=\) \(\ds W\) Union with Superset is Superset


By definition of meet-irreducible open:

$V \subseteq W$

Hence:

$y \in W$


Since $y$ was arbitrary:

$S \setminus \set x^- \subseteq W$


By definition of set equivalence:

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

$\blacksquare$


Also see


Sources

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