In topology and related branches of mathematics, a connected space is a topological space which cannot be written as the disjoint union of two or more nonempty spaces. 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. Topology (Greek topos, place and logos, study) is a branch of mathematics concerned with spatial properties preserved under bicontinuous deformation (stretching without tearing or gluing); these are the topological invariants. ... Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations related to: Mathematics Look up Mathematics on Wiktionary, the free dictionary Wikimedia Commons has media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In the mathematical field of topology a direct sum, direct disjoint sum or coproduct is an important universal construction for topological spaces. ... In topology and related areas of mathematics a topological property or topological invariant is a property of a topological space which is invariant under homeomorphisms. ... In mathematics, a path in a topological space X is a continuous map f from the unit interval I = [0,1] to X f : I → X. The initial point of the path is f(0) and the terminal point is f(1). ...

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. The space is not connected since two rectangles are disjoin. Another good example is a space with an annulus removed. The space is not connected since you cannot connect two points, one inside the annulus and the other outside; hence the term "connect".

Also, in a sense, a connected space is a generalization of an interval on the real number line, just as a topological space is, so to speak, an attempt to generalize an interval.

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. Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In set theory and other branches of mathematics, the union of a collection of sets is the set that contains everything that belongs to any of the sets, but nothing else. ... In mathematics, two sets are said to be disjoint if they have no element in common. ... In set theory, a set is called non-empty (or nonempty) if it contains at least one element, and is therefore not the empty set. ... In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U... A is a subset of B, and B is a superset of A. In mathematics, especially in set theory, a set A is a subset of a set B, if A is contained inside B. Every set is a subset of itself. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In mathematics and more specifically set theory, the empty set is the unique set which contains no elements. ...

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 (This follows since the complement of an open set is closed).
3. The only sets which are both open and closed (clopen sets) are X and the empty set.
4. The only sets with empty boundary are X and the empty set.
5. 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. In topology and related branches of mathematics, a closed set is a set whose complement is open. ... In set theory and other branches of mathematics, two kinds of complements are defined, the relative complement and the absolute complement. ... In topology, a clopen set (or closed-open set) in a topological space is a set which is both open and closed. ... In topology, the boundary of a subset S of a topological space X is the sets closure minus its interior. ... In topology and related branches of mathematics, separated sets are pairs of subsets of a given topological space that are related to each other in a certain way. ... In mathematics, especially in order theory, a maximal element of a subset S of some partially ordered set is an element of S that is not smaller than any other element in S. The term minimal element is defined dually. ... A partition of U into 6 blocks: a Venn diagram representation. ... In mathematics, two sets are said to be disjoint if they have no element in common. ... In topology and related branches of mathematics, a closed set is a set whose complement is open. ... In mathematics, a rational number (or informally fraction) is a ratio or quotient of two integers, usually written as the vulgar fraction a/b, where b is not zero. ... This is a glossary of some terms used in the branch of mathematics known as topology. ...

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.

In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ... In topology and related fields of mathematics, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points are isolated from each other in a certain sense. ... The Cantor set, introduced by German mathematician Georg Cantor, is a remarkable construction involving only the real numbers between zero and one. ...

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.) In topology and related areas of mathematics a continuous function is a morphism between topological spaces; that is, a mapping which preserves the topological structure. ... In mathematics, the unit interval is the interval [0,1], that is the set of all real numbers x such that zero is less than or equal to x and x is less than or equal to one. ... In mathematics, a path in a topological space X is a continuous map f from the unit interval I = [0,1] to X f : I → X. The initial point of the path is f(0) and the terminal point is f(1). ...

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. In topology, the long line is a topological space analogous to the real line, but much longer. ... In the branch of mathematics known as topology, the topologists sine curve is an example that has several interesting properties. ...

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. In mathematics, the real line is simply the set of real numbers. ... ↔ ⇔ ≡ logical symbols representing iff. ... In mathematics, interval is a concept relating to the sequence and set-membership of one or more numbers. ... In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U... In mathematics, a set is called finite if and only if there is a bijection between the set and some set of the form {1, 2, ..., n} where is a natural number. ...

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. In the mathematical field of topology a homeomorphism or topological isomorphism (from the Greek words homeos = identical and morphe = shape) is a special isomorphism between topological spaces which respects topological properties. ... In topology and related branches of mathematics, a Hausdorff space is a topological space in which points can be separated by neighbourhoods. ... In mathematics, especially order theory, a partially ordered set (or poset for short) is a set equipped with a partial order relation. ... In mathematics, a base (or basis) B for a topological space X with topology T is a collection of open sets in T such that every open set in T can be written as a union of elements of B. We say that the base generates the topology T. Bases... The title given to this article is incorrect due to technical limitations. ... In topology and related branches of mathematics, a Hausdorff space is a topological space in which points can be separated by neighbourhoods. ...

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. In mathematics, a base (or basis) B for a topological space X with topology T is a collection of open sets in T such that every open set in T can be written as a union of elements of B. We say that the base generates the topology T. Bases...

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 manifold M is a type of space, characterized in one of two equivalent ways: near every point of the space, we have a coordinate system; or near every point, the environment is like that in Euclidean space of a given dimension. ...

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.
• If is a family of connected subsets of a topological space X such that is nonempty for all i, 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 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.

In topology, a continuous function is generally defined as one for which preimages of open sets are open. ... In mathematics, the image of an element x in a set X under the function f : X → Y, denoted by f(x), is the unique y in Y that is associated with x. ... In analysis, the intermediate value theorem is either of two theorems of which an account is given below. ... ↔ ⇔ ≡ For other possible meanings of iff, see IFF. In mathematics, philosophy, logic and technical fields that depend on them, iff is used as an abbreviation for if and only if. Common alternative phrases to iff or if and only if include Q is necessary and sufficient for P and P... In mathematics, the closure of a set S consists of all points which are intuitively close to S. A point which is in the closure of S is a point of closure of S. The notion of closure is in many ways dual to the notion of interior. ... In topology and related areas of mathematics, a quotient space (also called an identification space) is, intuitively speaking, the result of identifying or gluing together certain points of a given space. ... This page is about a higher mathematics topic. ...

See also

• uniformly connected space
• connected component (graph theory)
• separated sets
• simply connected