# nLab irreducible topological space

A topological space $X$ is irreducible if it can not be expressed as union of two proper closed subsets, or equivalently if any two inhabited open subsets have inhabited intersection. A subset $S$ of a topological space $X$ is an irreducible subset if $S$ is an irreducible topological space with the induced topology. 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 April 9, 2015 16:41:49 by A. Gagna? (82.170.147.99)