Connected (topology)

Definition

A topological space [math](X,\mathcal{J})[/math] is connected if there is no separation of [math]X[/math]

Separation

This belongs on this page because a separation is only useful in this definition.

A separation of [math]X[/math] is a pair of two non-empty open sets [math]U,V[/math] where [math]U\cap V=\emptyset[/math] where [math]U\cup V=X[/math]

Equivalent definition

We can also say: A topological space [math](X,\mathcal{J})[/math] is connected if and only if the sets [math]X,\emptyset[/math] are the only two sets that are both open and closed.

Proof

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TODO:

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