PlanetMath: irreducible

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irreducible

(Definition)

A subset of a topological space is reducible if it can be written as a union $F = F_1 \cup F_2$ of two closed proper subsets , of (closed in the subspace topology). That is, is reducible if it can be written as a union $F = (G_1\cap F)\cup(G_2\cap F)$ where , are closed subsets of , neither of which contains .

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

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

"irreducible" is owned by mathcam. [ full author list (4) | owner history (3) ]

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Also defines:

reducible

Keywords:

topological space

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Cross-references: lines, hyperconnected, contains, closed subsets, subspace topology, proper subsets, closed, union, topological space, subset
There are 26 references to this entry.

This is version 10 of irreducible, born on 2001-12-20, modified 2006-02-17.
Object id is 1109, canonical name is IrreducibleClosedSet.
Accessed 5948 times total.

Classification:

AMS MSC:	54B05 (General topology :: Basic constructions :: Subspaces)
	14A10 (Algebraic geometry :: Foundations :: Varieties and morphisms)
	14A15 (Algebraic geometry :: Foundations :: Schemes and morphisms)

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