Disconnected (topology)/Definition

Grade: A

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Mendelson and Lee's topological manifolds have it covered, I think Munkres is where I got "separation" from

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].