topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ho ...

and related branches of mathematics, a connected space is a

topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...

that cannot be represented as the

union Union commonly refers to: * Trade union, an organization of workers * Union (set theory), in mathematics, a fundamental operation on sets Union may also refer to: Arts and entertainment Music * Union (band), an American rock group ** ''Un ...

of two or more

disjoint Disjoint may refer to: *Disjoint sets, sets with no common elements *Mutual exclusivity, the impossibility of a pair of propositions both being true See also *Disjoint union *Disjoint-set data structure {{disambig : Let

X

and

Y

be topological spaces and let

f:X\rightarrow 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 In mathematical analysis, the intermediate value theorem states that if f is a continuous function whose domain contains the interval , then it takes on any given value between f(a) and f(b) at some point within the interval. This has two im ...

. *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. Furthermore, any subset between a connected subset and its closure 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 In arithmetic, a quotient (from lat, quotiens 'how many times', pronounced ) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics, and is commonly referred to as the integer part of a ...

of a connected (resp. locally connected, path-connected, locally path-connected) space is connected (resp. locally connected, path-connected, locally path-connected). *Every

product Product may refer to: Business * Product (business), an item that serves as a solution to a specific consumer problem. * Product (project management), a deliverable or set of deliverables that contribute to a business solution Mathematics * Prod ...

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 In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a ...

is locally path-connected. *Arc-wise connected space is path connected, but path-wise connected space may not be arc-wise connected *Continuous image of arc-wise connected set is arc-wise connected.

Graphs

Graph Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discre ...

s 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 . 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

s, for instance: * If there exist no two disjoint non-empty open sets in a topological space

X

X

must be connected, and thus

hyperconnected space In the mathematical field of topology, a hyperconnected space or irreducible space is a topological space ''X'' that cannot be written as the union of two proper closed sets (whether disjoint or non-disjoint). The name ''irreducible space'' is pre ...

s are also connected. * Since a

simply connected space In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the spac ...

is, by definition, also required to be path connected, any simply connected space is also connected. 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 In mathematics, a topological space ''X'' is contractible if the identity map on ''X'' is null-homotopic, i.e. if it is homotopic to some constant map. Intuitively, a contractible space is one that can be continuously shrunk to a point within ...

. Every contractible space is path connected and thus also connected. In general, 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 In the branch of mathematics known as topology, the topologist's sine curve or Warsaw sine curve is a topological space with several interesting properties that make it an important textbook example. It can be defined as the graph of the functi ...

Graphs

Stronger forms of connectedness

See also

References

Further reading