Topological Closure of Singleton is Irreducible/Proof 1

Theorem

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

Let $x$ be a point of $T$.

Then:

$\set x^-$ is irreducible

where $\set x^-$ denotes the topological closure of $\set x$.

Proof

Follows from:

Trivial Topological Space is Irreducible

Closure of Irreducible Subspace is Irreducible

$\blacksquare$