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 specifically exclude the empty set with its unique topology as a connected space, but this encyclopedia does not follow this practice.
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.
A space X is said to be arc-connected if any two distinct points can be joined by an arc, that is a path f which is a homeomorphism between the unit interval [0,1] and its image f([0,1]). It can be shown any Hausdorff space which is path-connected is also arc-connected. An example of a space which is path-connected but not arc-connected is provided by adding a second copy 0' of 0 to the nonnegative real numbers [0,∞). One endows this set with a partial order by specifying that 0'<a for any positive number a, but leaving 0 and 0' incomparable. One then endows this set with the order topology, that is one takes the open intervals (a,b)={x | a<x<b} and the half-open intervals [0,a)={x | 0≤x<a}, [0',a)={x | 0'≤x<a} as a base for the topology. The resulting space is a T1 space but not a Hausdorff space. Clearly 0 and 0' can be connected by a path but not by an arc in this space.
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.
See also: Simply connected
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