Difference between revisions of "Disconnected (topology)"
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Revision as of 22:13, 30 September 2016
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- Note: much more information may be found on the connected page, this page exists just to document disconnectedness.
Contents
Definition
A topological space, [ilmath](X,\mathcal{ J })[/ilmath], is said to be disconnected if[1]:
- [ilmath]\exists U,V\in\mathcal{J}[U\ne\emptyset\wedge V\neq\emptyset\wedge V\cap U=\emptyset\wedge U\cup V=X][/ilmath], in words "if there exists a pair of disjoint and non-empty open sets, [ilmath]U[/ilmath] and [ilmath]V[/ilmath], such that their union is [ilmath]X[/ilmath]"
In this case, [ilmath]U[/ilmath] and [ilmath]V[/ilmath] are said to disconnect [ilmath]X[/ilmath][1] and are sometimes called a separation of [ilmath]X[/ilmath].
See also
- Connected - a space is connected if it is not disconnected
- Much more information is available on that page, this is simply a supporting page
References
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