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)