T1/2 Space is T0 Space
Theorem
Let $T = \struct {S, \tau}$ be a $T_{\frac 1 2}$ topological space.
Then $T$ is $T_0$ space.
Proof
By Characterization of T0 Space by Closures of Singletons it suffices to prove that
- $\forall x, y \in S: x \ne y \implies x \notin \set y^- \lor y \notin \set x^-$
where $\set y^-$ denotes the closure of $\set y$.
Let $x, y$ be points of $T$ such that:
- $x \ne y$
Aiming for a contradiction, suppose:
- $x \in \set y^- \land y \in \set x^-$
We will prove that:
- $x \notin \set x'$
where $\set x'$ denotes the derivative of $\set x$.
Aiming for a contradiction, suppose:
- $x \in \set x'$
As $S$ is open by definition of topological space by Characterization of Derivative by Open Sets:
- $\exists z \in S: z \in \set x \cap S \land z \ne x$
By definition of intersection:
- $z \in \set x$
This contradicts $z \ne x$ by definition of singleton.
Thus $x \notin \set x'$.
We will prove that:
- $(1): \quad \lnot \forall G \in \tau: y \in G \implies \set x \cap G \ne \O$
Aiming for a contradiction, suppose:
- $\forall G \in \tau: y \in G \implies \set x \cap G \ne \O$
As sublemma we will show that:
- $\forall U \in \tau: y \in U \implies \exists r \in S: r \in \set x \cap U \land y \ne r$
Let $U \in \tau$ such that:
- $y \in U$
Then by assumption:
- $\set x \cap U \ne \O$
By definition of empty set:
- $\exists z: z \in \set x \cap U$
By definition of intersection:
- $z \in \set x$
Then by definition of singleton:
- $z = x \ne y$
Thus:
- $\exists r \in S: r \in \set x \cap U \land y \ne r$
Then by Characterization of Derivative by Open Sets:
- $y \in \set x'$
By definition of relative complement:
- $y \notin \relcomp S {\set x'} \land x \in \relcomp S {\set x'}$
By definition of $T_{\frac 1 2}$ space:
- $\set x'$ is closed
By definition of closed set:
- $\relcomp S {\set x'}$ is open
By $x \in \set y^-$ and by Condition for Point being in Closure:
- $\set y \cap \relcomp S {\set x'} \ne \O$
Then by definition of empty set:
- $\exists z: z \in \set y \cap \relcomp S {\set x'}$
By definition of intersection:
- $z \in \set y \land z \in \relcomp S {\set x'}$
By definition of singleton:
- $z = y$
Thus $z \in \relcomp S {\set x'}$ contradicts $y \notin \relcomp S {\set x'}$.
This ends the proof of $(1)$.
Then by $(1)$ and Condition for Point being in Closure:
- $y \notin \set x^-$
This contradicts $y \in \set x^-$.
$\blacksquare$
Sources
- Mizar article TOPGEN_4:47