Path Connectedness
A path from a point x to a point y in a topological space X is a continuous function f from the unit interval to X with f(0) = x and f(1) = y. A path-component of X is an equivalence class of X under the equivalence relation which makes x equivalent to y if there is a path from x to y. The space X is said to be path-connected (or pathwise connected or 0-connected) if there is at most one path-component, i.e. if there is a path joining any two points in X. Again, many others exclude the empty space.
Every path-connected space is connected. The converse is not always true: examples of connected spaces that are not path-connected 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 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.
Read more about this topic: Connected Space
Famous quotes containing the word path:
“What is the use of going right over the old track again? There is an adder in the path which your own feet have worn. You must make tracks into the Unknown.”
—Henry David Thoreau (1817–1862)