 Connected space

For other uses, see Connection (disambiguation).
In topology and related branches of mathematics, a connected space is a topological space that 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 pathconnected 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.
As an example of a space that is not connected, one can delete an infinite line from the plane. Other examples of disconnected spaces (that is, spaces which are not connected) include the plane with a closed annulus removed, as well as the union of two disjoint open disks in twodimensional Euclidean space.
Contents
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 exclude the empty set (with its unique topology) as a connected space, but this article 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 a nonempty topological space are called the connected components of the space. The components of any topological space X form a partition of X: they are disjoint, nonempty, and their union is the whole space. (Since we are insisting on calling the empty topological space connected, we need a special convention here: the empty space has no connected components.) Every component is a closed subset of the original space. It follows that, in the case where their number is finite, each component is also an open subset. However, if their number is infinite, this might not be the case; for instance, the connected components of the set of the rational numbers are the onepoint sets, which are not open.
Let Γ_{x} be a connected component of x in a topological space X, and Γ_{x}' be the intersection of all openclosed sets containing x (called quasicomponent of x.) Then where the equality holds if X is compact Hausdorff or locally connected.
Disconnected spaces
A space in which all components are onepoint 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 threedimensional Euclidean space without the origin is connected, and even simply connected. In contrast, the onedimensional Euclidean space without the origin is not connected.
 A Euclidean plane with a straight line removed is not connected since it consists of two halfplanes.
 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 simplest example is the discrete twopoint space.^{[1]}
 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 (that is, the group of nbyn 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, 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 if and only if the ring is not a product of two rings in a nontrivial way.
Path connectedness
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 pathcomponent of X is an equivalence class of X under the equivalence relation which makes x equivalent to y if there is a path from x to y. The space X is said to be pathconnected (or pathwise connected or 0connected) if there is at most one pathcomponent, i.e. if there is a path joining any two points in X. Again, many others exclude the empty space.
Every pathconnected space is connected. The converse is not always true: examples of connected spaces that are not pathconnected 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 pathconnected; these subsets are the intervals of R. Also, open subsets of R^{n} or C^{n} are connected if and only if they are pathconnected. Additionally, connectedness and pathconnectedness are the same for finite topological spaces.
Arc connectedness
A space X is said to be arcconnected 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 pathconnected is also arcconnected. An example of a space which is pathconnected but not arcconnected 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 halfopen intervals [0, a) = {x  0 ≤ x < a}, [0', a) = {x  0' ≤ x < a} as a base for the topology. The resulting space is a T_{1} 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
Main article: Locally connected spaceA 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 pathconnected if it has a base of pathconnected sets. An open subset of a locally pathconnected space is connected if and only if it is pathconnected. This generalizes the earlier statement about R^{n} and C^{n}, each of which is locally pathconnected. More generally, any topological manifold is locally pathconnected.
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 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, is nonempty, then is also connected.
 If {A_{α}} is a nonempty family of connected subsets of a topological space X such that is nonempty, then is also connected.
 Every pathconnected space is connected.
 Every locally pathconnected space is locally connected.
 A locally pathconnected space is pathconnected 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 pathconnected components (which in general are neither open nor closed).
 Every quotient of a connected (resp. pathconnected) space is connected (resp. pathconnected).
 Every product of a family of connected (resp. pathconnected) spaces is connected (resp. pathconnected).
 Every open subset of a locally connected (resp. locally pathconnected) space is locally connected (resp. locally pathconnected).
 Every manifold is locally pathconnected.
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 5cycle graph (and any ncycle 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 forms of connectedness
There are stronger forms of connectedness for topological spaces, for instance:
 If there exist no two disjoint nonempty 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 does not need to 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, as does the above mentioned topologist's sine curve.
See also
 uniformly connected space
 locally connected space
 connected component (graph theory)
 nconnected
 Connectedness locus
 Extremally disconnected space
References
Notes
General references
 Munkres, James R. (2000). Topology, Second Edition. Prentice Hall. ISBN 0131816292.
 Weisstein, Eric W., "Connected Set" from MathWorld.
 V. I. Malykhin (2001), "Connected space", in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Springer, ISBN 9781556080104, 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.shimaneu.ac.jp/memoir/39/D.Buhagiar.pdf.
Categories: General topology
 Properties of topological spaces
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