irreducible
A subset F of a topological space X is reducible if it can be written as a union F=F1∪F2 of two closed proper subsets
F1, F2 of F (closed in the subspace topology). That is, F is reducible if it can be written as a union F=(G1∩F)∪(G2∩F) 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)∈ℝ2:xy=0} with the subspace topology from ℝ2. This space is a union of two lines {(x,y)∈ℝ2:x=0} and {(x,y)∈ℝ2:y=0}, which are proper closed subsets. So this space is reducible, and thus not irreducible.
Title | irreducible |
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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 |