A subset $F$ of a topological space $X$ is reducible if it can be written as a union $F = F_1 \cup F_2$ of two closed proper subsets $F_1$ $F_2$ of $F$ (closed in the subspace topology). That is, $F$ is reducible if it can be written as a union $F = (G_1\cap F)\cup(G_2\cap F)$ where $G_1$ $G_2$ are closed subsets of $X$ neither of which contains $F$

A subset of a topological space is irreducible (or hyperconnected) if it is not reducible.

As an example, consider $\{ (x,y)\in\mathbb{R}^2 : xy = 0 \}$ with the subspace topology from $\mathbb{R}^2$ This space is a union of two lines $\{ (x,y)\in\mathbb{R}^2 : x = 0 \}$ and $\{ (x,y)\in\mathbb{R}^2 : y = 0 \}$ which are proper closed subsets. So this space is reducible, and thus not irreducible.