Connected Space April 13, 2025 July 13, 2025 A topological space \(X\) is said to be connected if it is not disconnected . Remarks The only open and closed subsets are \(X\) and \(\emptyset\). This even characterizes connectedness. If \(A\subseteq U \dot{\cup}V\) is connected, then either \(A\subseteq U\) or \(A\subseteq V\). A connected dense subset implies connectedness. A path-connected space is connected. Examples Connected sets in \(\mathbb{R}\) are intervals. Sphere Torus New Spaces from Old Link to heading Closure of connected subsets are connected and every set lying between them. The union of connected subsets with one intersecting point are connected. Products of connected spaces are connected. Quotient spaces of connected spaces are connected. Double of a connected manifold is connected. See also Link to heading main theorem of connectedness intermediate value theorem component locally connected space