Connected space
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In topology and related branches of mathematics, a topological space X is said to be disconnected if it is the union of two disjoint nonempty open sets. Otherwise, X is said to be connected. A subset of a topological space is said to be connected if it is connected under its subspace topology. Some authors specifically exclude the empty set with its unique topology as a connected space, but this encyclopedia does not follow that practice.
For a topological space X the following conditions are equivalent:
- X is connected.
- X cannot be divided into two disjoint nonempty closed sets (This follows since the complement of an open set is closed).
- The only sets which are both open and closed (clopen sets) are X and the empty set.
- The only sets with empty boundary are X and the empty set.
- X cannot be written as the union of two nonempty separated sets.
The maximal nonempty connected subsets of any topological space are called the connected 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.
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Examples
- The space of real numbers with the usual topology is connected.
- Every discrete topological space is totally disconnected.
- The Cantor set is totally disconnected.
Path connectedness
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 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.
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.
Local connectedness
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.
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.
- 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.