Totally disconnected implies T1

Statement
Any totally disconnected space is a T1 space.

Facts used

 * 1) uses::Closure of one-point subset implies irreducible
 * 2) uses::Irreducible implies connected

Proof
Given: A totally disconnected space $$X$$, a point $$x \in X$$.

To prove: $$\overline{\{ x \}} = \{ x \}$$.

Proof: We prove this by noting that the closure $$\overline{\{ x \}}$$ is irreducible by Fact (1), hence connected by Fact (2). Hence, because $$X$$ is totally disconnected, it must be the singleton subset $$\{ x \}$$.