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:
“Childhood is an adventure both for children and for their parents. There should be freedom to explore and joy in discovery. The important discoveries for both parents and children seldom come at the points where the path is smooth and straight. It is the curves in that path to adventure that make the trip interesting and worthwhile.”
—Lawrence Kutner (20th century)