nLab irreducible topological space

Contents

Context

Topology

topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

Contents

Definition

Definition

A topological space XX is called irreducible if it cannot be expressed as a finite union of proper closed subsets, or equivalently if any finite collection of inhabited open subsets has inhabited intersection.

If XX is irreducible according to Def. , then:

Remark

(role of the empty set)
Another common version of Def. (cf. Wikipedia) is to have it speak just about binary unions, as opposed to finite unions. With that definition the empty set would be irreducible, while according to Def. it is not: The empty set is the empty union and hence a finite union of proper closed subsets, but not a binary union of such, since it has no proper subsets!

But having the empty set count as reducible is useful, cf. Ex. below.

Definition

A subset SS of a topological space XX is an irreducible subset if SS is an irreducible topological space with the subspace topology.

Examples

Example

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

Example

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

References

See also:

Last revised on June 5, 2025 at 15:22:55. See the history of this page for a list of all contributions to it.