Connected space

In topology and related branches of mathematics, a connected space is a topological space which cannot be written as the disjoint union of two or more nonempty spaces. Connectedness is one of the principal topological properties that is used to distinguish topological spaces. A stronger notion is that of a path-connected space, which is a space where any two points can be joined by a path.

Theorems

• Main theorem: Let X and Y be topological spaces and let f : X → Y be a continuous function. If X is connected (resp. path-connected) then the image f(X) is connected (resp. path-connected). The intermediate value theorem can be considered as a special case of this result.
• If $\{A\_1,\; A\_2,ldots\}$ is a family of connected subsets of a topological space X such that $A\_i\; cap\; A\_\{i+1\}$ is nonempty for all i, then $cup\; A\_i$ 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 iff it is connected.
• The connected components of a space are disjoint unions of the path-connected components.
• The components of a locally connected space are open (and closed).
• The closure of a connected subset is connected.
• 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.