Connected space
From Academic Kids

In topology and related branches of mathematics, 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 (This follows since the complement of an open set is closed).
 The only sets which are both open and closed (clopen sets) are X and the empty set.
 The only sets with empty boundary are X and the empty set.
 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 onepoint sets. A space in which all components are onepoint 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.
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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.
Path connectedness
The space X is said to be pathconnected 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.)
Every pathconnected space is connected. Example 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.
A space X is said to be arcconnected 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
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.
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 connected (resp. pathconnected) then the image f(X) is connected (resp. pathconnected). The intermediate value theorem can be considered as a special case of this result.
 Every pathconnected space is connected.
 Every locally pathconnected space is locally connected.
 A locally pathconnected space is pathconnected iff it is connected.
 The connected components of a space are disjoint unions of the pathconnected 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. 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.