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A space is connected iff it cannot be divided into two disjoint nonempty closed sets (since the complement of an open set is closed). Furthermore, a space is connected iff its only clopen subsets are the empty set and the space itself.

The space X is said to be path-connected if for any two points x and y in X there exists a continuous functionIn mathematics, a function is a relation such that each element of a set (the domain is associated with a unique element of another (possibly the same) set (the codomain not to be confused with the range . The concept of a function is fundamental to virtu f from the unit intervalIn 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. The unit interval plays a fundamental role in homotopy theory, a major branch of t [0,1] to X with f(0) = x and f(1) = y. (This function is called a pathIn 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). One often speaks of a "path from x to y where x and y are the, or curveThis article is about the term used in mathematics. There is also a magazine called Curve. Metric geometry Geometry Topology General topology In mathematics, the concept of a curve tries to capture our intuitive idea of a geometrical one-dimensional and c, from x to y.)

Every path-connected space is connected. Example of connected spaces that are not path-connected include the extended long lineIn topology, the long line is a topological space analogous to the real line, but much longer. Because it behaves locally just like the real line, but has different large-scale properties, it serves as one of the basic counterexamples of topology. Definit L* and the topologist's sine curveIn the branch of mathematics known as topology, the topologist's sine curve is an example that has several interesting properties. It can be defined as a subset of the Euclidean plane as follows. Let S be the graph of the function sin(1/x) over the interv. The latter is a certain subset of the Euclidean plane:

{ (x,y) in R² | 0 < x and y = sin(1/x) } union { (0,y) in R² | -1 ≤ y ≤ 1 }.

However, subsets of the real lineIn mathematics, the real line is simply the set of real numbers. However, this term is usually used when R is to be treated as a space of some sort, such as a topological space or a vector space. The real line has been studied at least since the days of t 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.

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 | abase 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.

If X and Y are topological spaces, f is a continuous function from X to Y, and X is connected (respectively, path-connected), then the image f(X) is connected (respectively, path-connected). The intermediate value theorem can be considered as a special case of this result.

The maximal nonempty connected subsets of any topological space are called the 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.

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 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.

See also: Simply connected