irreducible topological space

A topological space $X$ is called **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 subspace topology.

An algebraic variety is irreducible if its underlying topological space (in the Zariski topology) is irreducible.

A *sober topological space*, is one whose only irreducible closed subsets are the closures of single points.

Revised on April 5, 2017 14:18:44
by Urs Schreiber
(78.47.168.108)