nLab irreducible topological space




A topological space XX 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 SS of a topological space XX is an irreducible subset if SS 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.

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