Disconnected (topology)

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Note: much more information may be found on the connected page, this page exists just to document disconnectedness.


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].

Disconnected subset

Let [ilmath]A\in\mathcal{P}(X)[/ilmath] be an arbitrary subset of [ilmath]X[/ilmath] (for a topological space [ilmath](X,\mathcal{ J })[/ilmath] as given above), then we say [ilmath]A[/ilmath] is disconnected in [ilmath](X,\mathcal{ J })[/ilmath] if[2]:

Equivalent conditions

To a topological space [ilmath](X,\mathcal{ J })[/ilmath] being connected:

To an arbitrary subset, [ilmath]A\in\mathcal{P}(X)[/ilmath], being connected:

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


  1. 1.0 1.1 Introduction to Topological Manifolds - John M. Lee
  2. 2.0 2.1 Introduction to Topology - Bert Mendelson