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.