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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 . 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 disjoint. 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 Set s. 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 Set s (This follows since the Complement of an open set is closed). #The only sets which are both open and closed ( Clopen Set s) 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 Number s, for instance, are the one-point sets. A space in which all components are one-point sets is called totally disconnected. 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 Neighborhood s ''U'' of ''x'' and ''V'' of ''y'' such that ''X'' is the union of ''U'' and ''V''. 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, the space is not totally separated, or even Hausdorff . EXAMPLES
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 {Link without Title} to ''X'' with ''f''(0) = ''x'' and ''f''(1) = ''y''. (This function is called a '' Path '' from ''x'' to ''y''.) Every path-connected space is connected.
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 Subset s of R''n'' or '''C'''''n'' are connected if and only if they are path-connected. Additionally, connectedness and path-connectedness are the same for 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 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 |
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