# nLab irreducible topological space

A subset $S$ of a topological space $X$ is an irreducible subset if it can not be expressed as union of two proper closed subsets, or equivalently if any two inhabited open subsets have inhabited intersection. An irreducible topological space is a topological space which is an irreducible subset of itself. An algebraic variety is irreducible if its underlying topological space (in the Zariski topology) is irreducible.

Contrast this with a sober space, where the only irreducible closed subsets are the points.

Revised on August 4, 2009 20:11:48 by Toby Bartels (71.104.230.172)