= Closure of a point is irreducible
{title2=$\overline{\{x\}}$}
In any <topological space>, the <closure> $C$ of a point $x$ is an <irreducible closed subset>. If $C=A\cup B$ with $A,B$ closed in $C$, then both are <closed subsets> of the ambient <topological space>. One contains $x$, so it contains all of $C$ by the definition of <closure>. Equivalently, $x$ is a <generic point> of $C$. This does not assert that every <irreducible closed subset> of an arbitrary <topological space> has a <generic point>.
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