Connected space
In topology and related branches of mathematics, a topological space is said to be connected if it cannot be divided into two disjoint nonempty open sets whose union is the entire space. A subset of a topological space is said to be connected if it is connected carrying the subspace topology.
(Some authors do not consider the empty set (with its unique topology) as a connected space, but BambooWeb does.)
A space is connected iff it cannot be divided into two disjoint nonempty closed sets (since the complement of an open set is closed). Furthermore, a space is connected iff its only clopen subsets are the empty set and the space itself.
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 space X is totally disconnected iff, for any two elements x and y of X, there exist disjoint open neighborhoods U of x and V of y such that X is the union of U and V.
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 is not 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.
