A subset F of a topological spaceMathworldPlanetmath X is reducible if it can be written as a union F=F1F2 of two closed proper subsetsMathworldPlanetmathPlanetmath F1, F2 of F (closed in the subspace topology). That is, F is reducible if it can be written as a union F=(G1F)(G2F) where G1,G2 are closed subsets of X, neither of which contains F.

A subset of a topological space is irreducible (or hyperconnected) if it is not reducible.

As an example, consider {(x,y)2:xy=0} with the subspace topology from 2. This space is a union of two lines {(x,y)2:x=0} and {(x,y)2:y=0}, which are proper closed subsets. So this space is reducible, and thus not irreducible.

Title irreducible
Canonical name Irreducible1
Date of creation 2013-03-22 12:03:30
Last modified on 2013-03-22 12:03:30
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 14
Author mathcam (2727)
Entry type Definition
Classification msc 14A15
Classification msc 14A10
Classification msc 54B05
Related topic IrreducibleComponent
Related topic HyperconnectedSpace
Defines reducible