Closure of one-point subset implies irreducible

Statement
Suppose $$X$$ is a topological space and $$x$$ is a point in $$X$$. Let $$A$$ be the closure of $$\{ x \}$$ in $$X$$. Then, $$A$$ is an irreducible space with the subspace topology from $$X$$.

Related facts
A sober space is a topological space where the only irreducible closed subsets are the closures of one-point subsets. We have Hausdorff implies sober.