Connected space

Connected and disconnected subspaces of R². The space A at top is connected; the shaded space B at bottom is not.

In topology and related branches of mathematics, a connected space is a topological space which cannot be represented as the union of two or more disjoint nonempty open subsets. 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.

A subset of a topological space X is a connected set if it is a connected space when viewed as a subspace of X.

One may perceive mathematical spaces which are not connected. For instance, the space resulting from the deletion of an infinite line from the plane is not connected. Other examples of disconnected spaces (that is, spaces which are not connected) include the plane with an annulus removed, as well as the union of two disjoint disks in two-dimensional Euclidean space.

1 Formal definition
- 1.1 Connected components
- 1.2 Disconnected spaces
2 Examples
3 Path connectedness
4 Local connectedness
5 Theorems
6 Graphs
7 Stronger properties
8 See also
9 References

Formal definition

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.
The only subsets of X which are both open and closed (clopen sets) are X and the empty set.
The only subsets of X with empty boundary are X and the empty set.
X cannot be written as the union of two nonempty separated sets.
The only continuous functions from X to {0,1} are constant.

Connected components

The maximal 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.

Let $\Gamma_x$ be a connected component of x in a topological space X, and $\Gamma_x'$ be the intersection of all open-closed sets containing x (called quasi-component of x.) Then $\Gamma_x \subset \Gamma'_x$ where the equality holds if X is either compact Hausdorff or locally connected.

Disconnected spaces

A space in which all components are one-point sets is called totally disconnected. Related to this property, a space X is called totally separated if, 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. Clearly any totally separated space is totally disconnected, but the converse does not hold. For example take two copies of the rational numbers Q, and identify them at every point except zero. The resulting space, with the quotient topology, is totally disconnected. However, by considering the two copies of zero, one sees that the space is not totally separated. In fact, it is not even Hausdorff, and the condition of being totally separated is strictly stronger than the condition of being Hausdorff.

Examples

The closed interval [0, 2] in the standard subspace topology is connected; although it can, for example, be written as the union of [0, 1) and [1, 2], the second set is not open in the aforementioned topology of [0, 2].
The union of [0, 1) and (1, 2] is disconnected; both of these intervals are open in the standard topological space [0, 1) ∪ (1, 2].
(0, 1) ∪ {3} is disconnected.
A convex set is connected; it is actually simply connected.
A Euclidean plane excluding the origin, (0, 0), is connected, but is not simply connected. The three-dimensional Euclidean space without the origin is connected, and even simply connected. In contrast, the one-dimensional Euclidean space without the origin is not connected.
A Euclidean plane with a straight-line removed is not connected since it consists of two half-planes.
The space of real numbers with the usual topology is connected.
Any topological vector space over a connected field is connected.
Every discrete topological space with at least two elements is disconnected, in fact such a space is totally disconnected.
The Cantor set is totally disconnected; since the set contains uncountably many points, it has uncountably many components.
If a space X is homotopy equivalent to a connected space, then X is itself connected.
The topologist's sine curve is an example of a set that is connected but is neither path connected nor locally connected.
The general linear group $\operatorname{GL}(n, \mathbf{R})$ (that is, the group of n-by-n real matrices) consists of two connected components: the one with matrices of positive determinant and the other of negative determinant. In particular, it is not connected. In contrast, $\operatorname{GL}(n, \mathbf{C})$ is connected. More generally, the set of invertible bounded operators on a (complex) Hilbert space is connected.
The spectrum of a commutative local ring is connected. More generally, the spectrum of a commutative ring is connected if and only if it has no idempotent $\ne 0, 1$ if and only if the ring is not a product of two rings.

Path connectedness

This subspace of R² is path-connected, because a path can be drawn between any two points in the space.

A path from a point x to a point y in a topological space X is a continuous function f from the unit interval [0,1] to X with f(0) = x and f(1) = y. A path-component of X is an equivalence class of X under the equivalence relation defined by x is equivalent to y if there is a path from x to y. The space X is said to be path-connected (or pathwise connected or 0-connected) if there is only one path-component, i.e. if there is a path joining any two points in X.

Every path-connected space is connected. The converse is not always true: examples 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 or arcwise 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 at a point x if every neighbourhood of x contains a connected open neighbourhood. It is 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 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 (path-)connected then the image f(X) is (path-)connected. This result can be considered a generalization of the intermediate value theorem.
If $\{A_i\}_{i \in I}$ is a family of connected subsets of a topological space X indexed by an arbitrary set $I$ such that for all $i$ , $j$ in $I$ , $A_i \cap A_{j}$ is nonempty, then $\cup A_i$ is also connected.
If $\{A_\alpha\}$ is a nonempty family of connected subsets of a topological space X such that $\cap A_\alpha$ is nonempty, then $\cup A_\alpha$ 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 if and only if it is connected.
The closure of a connected subset is connected.
The connected components are always closed (but in general not open)
The connected components of a locally connected space are also open.
The connected components of a space are disjoint unions of the path-connected components (which in general are neither open nor closed).
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.

Graphs

Graphs have path connected subsets, namely those subsets for which every pair of points has a path of edges joining them. But it is not always possible to find a topology on the set of points which induces the same connected sets. The 5-cycle graph (and any n-cycle with n>3 odd) is one such example.

As a consequence, a notion of connectedness can be formulated independently of the topology on a space. To wit, there is a category of connective spaces consisting of sets with collections of connected subsets satisfying connectivity axioms; their morphisms are those functions which map connected sets to connected sets (Muscat & Buhagiar 2006). Topological spaces and graphs are special cases of connective spaces; indeed, the finite connective spaces are precisely the finite graphs.

However, every graph can be canonically made into a topological space, by treating vertices as points and edges as copies of the unit interval (see topological graph theory#Graphs as topological spaces). Then one can show that the graph is connected (in the graph theoretical sense) if and only if it is connected as a topological space.

Stronger properties

There are stronger forms of connectedness for a topological space. For instance, if there exist no two disjoint non-empty open sets in a topological space, X, X must be connected and thus hyperconnected spaces are also connected. Since a simply connected space is by definition, also required to be path connected, any simply connected space is also connected. Note however, that if the "path connectedness" requirement is dropped from the definition of simple connectivity, a simply connected space need not be connected.

Yet stronger versions of connectivity include the notion of a contractible space. Every contractible space is path connected and thus also connected. In general, note that any path connected space must be connected but there exist connected spaces that are not path connected. The deleted comb space furnishes such an example.

References

Munkres, James R. (2000). Topology, Second Edition. Prentice Hall. ISBN 0-13-181629-2.
Weisstein, Eric W., "Connected Set" from MathWorld.
V. I. Malykhin (2001), "Connected space", in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Springer, ISBN 978-1556080104, http://eom.springer.de/C/c025120.htm
Muscat, J; Buhagiar, D (2006). "Connective Spaces". Mem. Fac. Sci. Eng. Shimane Univ., Series B: Math. Sc. 39: 1–13. http://www.math.shimane-u.ac.jp/memoir/39/D.Buhagiar.pdf .