Connected space

A separation of \(X\) is a pair of disjoint nonempty open sets whose union is \(X\); the space is connected when it admits none. Equivalently, the only subsets that are both open and closed are \(\varnothing\) and \(X\), because if \(U\) is clopen, \(U\) and \(X\setminus U\) form a separation. A subset \(A\subseteq X\) is connected if it is connected as a subspace.

On the real line the connected sets are exactly the intervals. Hence \(\mathbb{R}\setminus\{0\}\) is not connected (it is split by \((-\infty,0)\) and \((0,\infty)\)) and neither is \(\mathbb{Q}\), which is separated at any irrational cut. Among finite spaces, the trivial topology is connected and the discrete one with two or more points is not.

Operations respect connectedness: the union of connected sets sharing a point is connected, and the continuous image of a connected space is connected. The second gives the intermediate value theorem: if \(f:[a,b]\to\mathbb{R}\) is continuous, \(f([a,b])\) is an interval. It also tells spaces apart, since a homeomorphism maps connected sets to connected sets.

Beware: connected is not the same as path-connected, although the latter implies the former. And do not read “connected” as “without holes” in the geometric sense: a circle is connected even though it encloses a hole.