Totally disconnected implies T1

This article gives the statement and possibly, proof, of an implication relation between two topological space properties. That is, it states that every topological space satisfying the first topological space property (i.e., totally disconnected space) must also satisfy the second topological space property (i.e., T1 space)
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Statement

Any totally disconnected space is a T1 space.

Facts used

1. Closure of one-point subset implies irreducible
2. 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 $\{x\}$.