A topological space is disconnected if and only if it is homeomorphic to a disjoint union of two or more non-empty topological spaces

Statement

Let [ilmath](X,\mathcal{ J })[/ilmath] be a topological space, then:

• [ilmath](X,\mathcal{ J })[/ilmath] is disconnected (ie: not connected) if and only if it is homeomorphic to a disjoint union of two or more non-empty topological spaces

Proof

Grade: C

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Exercise in Lee's top. manifolds, didn't take me very long to do, note that A topological space is disconnected if and only if there exists a non-constant continuous function from the space to the discrete space on two elements is a useful "precursor" theorem

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See also

• A topological space is disconnected if and only if there exists a non-constant continuous function from the space to the discrete space on two elements