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 disjoint union of two or more nonempty open subsets. Topology ( Greek topos, "place" and logos, "study" is the branch of Mathematics that studies the properties of Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. In General topology and related areas of Mathematics, the disjoint union (also called the direct sum, free union, or coproduct) of In Metric topology and related fields of Mathematics, a set U is called open if intuitively speaking starting from any point x in Connectedness is one of the principal topological properties that is used to distinguish topological spaces. In Topology and related areas of Mathematics a topological property or topological invariant is a property of a Topological space which is A stronger notion is that of a path-connected space, which is a space where any two points can be joined by a path. In Mathematics, a path in a Topological space X is a continuous map f from the Unit interval I = to

A subset of a topological space X is a connected set if it is a connected space when viewed as a subspace of X. In Topology and related areas of Mathematics, a subspace of a Topological space X is a Subset S of X which is

It is usually easy to think about what is not connected. A simple example would be a space consisting of two rectangles, each of which is a space and not adjoined to the other. In Geometry, a rectangle is defined as a Quadrilateral where all four of its angles are Right angles A rectangle with vertices ABCD would be denoted as The space is not connected since two rectangles are disjoint. Another good example is a space with an annulus removed. In Mathematics, an annulus (the Latin word for "little ring" with plural annuli) is a ring-shaped geometric figure or more generally a term The space is not connected since you cannot connect two points, one inside the annulus and the other outside; hence the term "connect".

Formal definition

A topological space X is said to be disconnected if it is the union of two disjoint nonempty open sets. Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. In Set theory, the term Union (denoted as ∪ refers to a set operation used in the convergence of set elements to form a resultant set containing the elements of both sets In Mathematics, and more specifically Set theory, the empty set is the unique set having no ( Zero) members In Metric topology and related fields of Mathematics, a set U is called open if intuitively speaking starting from any point x in 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. In Topology and related areas of Mathematics, a subspace of a Topological space X is a Subset S of X which is Some authors specifically exclude the empty set with its unique topology as a connected space, but this encyclopedia does not follow that practice. In Mathematics, and more specifically Set theory, the empty set is the unique set having no ( Zero) members

For a topological space X the following conditions are equivalent:

1. X is connected.
2. X cannot be divided into two disjoint nonempty closed sets. In Topology and related branches of Mathematics, a closed set is a set whose complement is open.
3. The only subsets of X which are both open and closed (clopen sets) are X and the empty set. In Topology, a clopen set (or closed-open set, a Portmanteau word in a Topological space is a set which is both open and closed
4. The only subsets of X with empty boundary are X and the empty set. For a different notion of boundary related to Manifolds see that article
5. X cannot be written as the union of two nonempty separated sets. In Topology and related branches of Mathematics, separated sets are pairs of Subsets of a given Topological space that are related to each other

The maximal connected subsets of any topological space are called the connected components of the space. In Mathematics, especially in Order theory, a maximal element of a subset S of some Partially ordered set is an element of S that The components form a partition of the space (that is, they are disjoint and their union is the whole space). In Mathematics, a partition of a set X is a division of X into non-overlapping " parts " or " blocks " Every component is a closed subset of the original space. In Topology and related branches of Mathematics, a closed set is a set whose complement is open. The components in general need not be open: the components of the rational numbers, for instance, are the one-point sets. In Mathematics, a rational number is a number which can be expressed as a Ratio of two Integers Non-integer rational numbers (commonly called fractions A space in which all components are one-point sets is called totally disconnected. In Topology and related branches of Mathematics, a totally disconnected space is a Topological space which is maximally disconnected in the sense that 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. In Topology and related areas of Mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a Topological space. 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. In Topology and related branches of Mathematics, a Hausdorff space, separated space or T2 space is a Topological space

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]. In Topology and related areas of Mathematics, a subspace of a Topological space X is a Subset S of X which is
• 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. In Euclidean space, an object is convex if for every pair of points within the object every point on the Straight line segment that joins them is also within the In Topology, a geometrical object or space is called simply connected (or 1-connected) if it is Path-connected and every path between two points can be
• 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. In Mathematics, the real numbers may be described informally in several different ways
• Any topological vector space over a connected field is connected. In Mathematics, a topological vector space is one of the basic structures investigated in Functional analysis.
• Every discrete topological space with at least two elements is disconnected, in fact such a space is totally disconnected. In Topology, a discrete space is a particularly simple example of a Topological space or similar structure one in which the points are " isolated " In Topology and related branches of Mathematics, a totally disconnected space is a Topological space which is maximally disconnected in the sense that
• The Cantor set is totally disconnected; since the set contains uncountably many points, it has uncountably many components. In Mathematics, the Cantor set, introduced by German Mathematician Georg Cantor in 1883 (but discovered in 1875 by Henry John Stephen Smith
• If a space X is homotopic to a connected space, then X is itself connected. In Topology, two continuous functions from one Topological space to another are called homotopic ( Greek homos = identical
• The topologist's sine curve is an example of a set that is connected but is neither path connected nor locally connected. In the branch of Mathematics known as Topology, the topologist's sine curve is a Topological space with several interesting properties that make it an important

Path connectedness

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

The space X is said to be path-connected (or pathwise connected or 0-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. In Topology and related areas of Mathematics a continuous function is a Morphism between Topological spaces Intuitively this is a function In Mathematics, the unit interval is the interval, that is the set of all Real numbers x such that zero is less than or equal to x (This function is called a path from x to y. In Mathematics, a path in a Topological space X is a continuous map f from the Unit interval I = to )

Every path-connected space is connected. The reverse 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. In Topology, the long line (or Alexandroff line) is a Topological space analogous to the Real line, but much longer In the branch of Mathematics known as Topology, the topologist's sine curve is a Topological space with several interesting properties that make it an important

However, subsets of the real line R are connected if and only if they are path-connected; these subsets are the intervals of R. In Mathematics, the real line is simply the set R of singleton Real numbers However this term is usually used when R is to be treated as a ↔ In Mathematics, an interval is a set of Real numbers with the property that any number that lies between two numbers in the set is also included in the set Also, open subsets of Rⁿ or Cⁿ are connected if and only if they are path-connected. In Metric topology and related fields of Mathematics, a set U is called open if intuitively speaking starting from any point x in 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]). Topological equivalence redirects here see also Topological equivalence (dynamical systems. It can be shown any Hausdorff space which is path-connected is also arc-connected. In Topology and related branches of Mathematics, a Hausdorff space, separated space or T2 space is a Topological space 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. In Mathematics, especially Order theory, a partially ordered set (or poset) formalizes the intuitive concept of an ordering sequencing or arrangement 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. In Mathematics, a base (or basis) B for a Topological space X with topology T is a collection of Open sets The resulting space is a T₁ space but not a Hausdorff space. In Topology and related branches of Mathematics, T1 spaces and R0 spaces are particular kinds of Topological spaces The In Topology and related branches of Mathematics, a Hausdorff space, separated space or T2 space is a Topological 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 space

A topological space is said to be locally connected at a point x if every neighbourhood of x contains a connected open neighbourhood. In Topology and other branches of Mathematics, a Topological space is said to be locally connected at x (where x is a point In Topology and other branches of Mathematics, a Topological space is said to be locally connected at x (where x is a point It is locally connected if it has a base of connected sets. In Mathematics, a base (or basis) B for a Topological space X with topology T is a collection of Open 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. In the branch of Mathematics known as Topology, the topologist's sine curve is a Topological space with several interesting properties that make it an important

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. In Mathematics, a topological manifold is a Hausdorff Topological space which looks locally like Euclidean space in a sense defined below

Theorems

• Main theorem: Let X and Y be topological spaces and let f : X → Y be a continuous function. In Topology and related areas of Mathematics a continuous function is a Morphism between Topological spaces Intuitively this is a function If X is connected (resp. path-connected) then the image f(X) is connected (resp. In Mathematics, the image of a preimage under a given function is the set of all possible function outputs when taking each element of the preimage path-connected). This result can be considered a generalization of the intermediate value theorem. In Mathematical analysis, the intermediate value theorem states that for each value between the upper and lower bounds of the image of a Continuous function
• 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 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 closure of a connected subset is connected. In Mathematics, the closure of a set S consists of all points which are intuitively "close to S "
• The connected components are always closed (but in general not open)
• The connected components of a locally connected space are also open. In Topology and related branches of Mathematics, a closed set is a set whose complement is 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. In Topology and related areas of Mathematics, a quotient space (also called an identification space) is intuitively speaking the result of identifying path-connected) space is connected (resp. path-connected).
• Every product of a family of connected (resp. In Topology and related areas of Mathematics, a product space is the Cartesian product of a family of Topological spaces equipped with a natural 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. A manifold is a mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be

Graphs

Graphs have path connected subsets, namely those subsets for which every pair of points has a path of edges joining them. In Mathematics and Computer science, a graph is the basic object of study in Graph theory. But not every graph has a topology which induces the same connected sets. The 5-cycle graph (and any n-cycle with n>3 odd) is one such example. In Graph theory, a cycle graph is a graph that consists of a single cycle, or in other words some number of vertices connected in a closed chain

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.

See also

• locally connected space
• uniformly connected space
• connected component (graph theory)
• simply connected space
• n-connected
• hyperconnected space

References

• Munkres, James R. (2000). In Topology and other branches of Mathematics, a Topological space is said to be locally connected at x (where x is a point In Topology and related areas of Mathematics a uniformly connected space or Cantor connected space is a Uniform space U such that In an Undirected graph, a connected component or component is a maximal connected subgraph In Topology, a geometrical object or space is called simply connected (or 1-connected) if it is Path-connected and every path between two points can be n-connected redirects here for the concept in graph theory see Connectivity (graph theory. In Mathematics, a hyperconnected space is a Topological space X that cannot be written as the union of two proper closed sets James Raymond Munkres is a Professor Emeritus of Mathematics at MIT and the author of several texts in the area of Topology, including Topology Topology, Second Edition. Prentice Hall. ISBN 0-13-181629-2.  
• Eric W. Weisstein, Connected Set at MathWorld. Eric W Weisstein (born March 18, 1969, in Bloomington Indiana) is an Encyclopedist who created and maintains MathWorld MathWorld is an online Mathematics reference work created and largely written by Eric W
• V. I. Malykhin (2001), “Connected space”, in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104 
• Muscat, J & Buhagiar, D (2006), “Connective Spaces”, Mem. The Encyclopaedia of Mathematics is a large reference work in Mathematics. Fac. Sci. Eng. Shimane Univ. , Series B: Math. Sc. 39: 1-13, <http://www.math.shimane-u.ac.jp/common/memoir/39/D.Buhagiar.pdf> .