irreducibleIrreducibleClosedSet2001-12-20 01:41:562006-02-17 11:23:10Definitionreducibletopological space% this is the default PlanetMath preamble. as your knowledge
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% define commands hereA subset $F$ of a topological space $X$ is {\em 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 {\em irreducible} (or \emph{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.