Connected Space - Theorems

Theorems

• Main theorem: Let X and Y be topological spaces and let f : X → Y be a continuous function. If X is (path-)connected then the image f(X) is (path-)connected. This result can be considered a generalization of the intermediate value theorem.
• If is a family of connected subsets of a topological space X indexed by an arbitrary set such that for all, in, is nonempty, then is also connected.
• If is a nonempty family of connected subsets of a topological space X such that is nonempty, then is also connected.
• Every path-connected space is connected.
• Every locally path-connected space is locally connected.
• A locally path-connected space is path-connected if and only if it is connected.
• The closure of a connected subset is connected.
• The connected components are always closed (but in general not open)
• The connected components of a locally connected space are also open.
• The connected components of a space are disjoint unions of the path-connected components (which in general are neither open nor closed).
• Every quotient of a connected (resp. path-connected) space is connected (resp. path-connected).
• Every product of a family of connected (resp. path-connected) spaces is connected (resp. path-connected).
• Every open subset of a locally connected (resp. locally path-connected) space is locally connected (resp. locally path-connected).
• Every manifold is locally path-connected.