# Disconnected (topology)

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## Definition

A topological space, [ilmath](X,\mathcal{ J })[/ilmath], is said to be disconnected if:

• [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] 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:

## 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: