nLab irreducible topological space

Definition

A topological space $X$ is called irreducible if it cannot 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.

Related concepts

• separation axiom