# irreducible

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.

Title irreducible Irreducible1 2013-03-22 12:03:30 2013-03-22 12:03:30 mathcam (2727) mathcam (2727) 14 mathcam (2727) Definition msc 14A15 msc 14A10 msc 54B05 IrreducibleComponent HyperconnectedSpace reducible