# A topological space is connected if and only if the only sets that are both open and closed in the space are the entire space itself and the emptyset

## Statement

Let [ilmath](X,\mathcal{ J })[/ilmath] be a topological space, then[1][2]:

• [ilmath](X,\mathcal{ J })[/ilmath] is connected if and only if the only two sets that are both open and closed in [ilmath](X,\mathcal{ J })[/ilmath] are [ilmath]X[/ilmath] itself and [ilmath]\emptyset[/ilmath]