Connected space : Connected (topology) 

The space X is said to be path-connected if for any two points x and y in X there exists a continuous function f from the unit interval [0,1] to X with f(0) = x and f(1) = y. (This function is called a path, or curve, from x to y.)

Every path-connected space is connected. Example of connected spaces that are not path-connected include the extended long line L* and the topologist's sine curve[?]. The latter is a certain subset of the Euclidean plane:

{ (x,y) in R² | 0 < x and y = sin(1/x) } union { (0,y) in R² | -1 ≤ y ≤ 1 }.

However, subsets of the real line R are connected if and only if they are path-connected; these subsets are the intervals of R. Also, open subsets of Rⁿ or Cⁿ are connected if and only if they are path-connected. Additionally, connectedness and path-connectedness are the same for finite topological spaces.

If X and Y are topological spaces, f is a continuous function from X to Y, and X is connected (respectively, path-connected), then the image f(X) is connected (respectively, path-connected). The intermediate value theorem can be considered as a special case of this result.

The maximal[?] nonempty connected subsets of any topological space are called the components of the space. The components form a partition of the space (that is, they are disjoint and their union is the whole space). Every component is a closed subset of the original space. The components in general need not be open: the components of the rational numbers, for instance, are the one-point sets. A space in which all components are one-point sets is called totally disconnected.

A topological space is said to be locally connected if it has a base of connected sets. It can be shown that a space X is locally connected if and only if every component of every open set of X is open. The topologist's sine curve shown above is an example of a connected space that isn't locally connected.

Similarly, a topological space is said to be locally path-connected if it has a base of path-connected sets. An open subset of a locally path-connected space is connected if and only if it is path-connected. This generalizes the earlier statement about Rⁿ and Cⁿ, each of which is locally path-connected. More generally, any topological manifold is locally path-connected.

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