Topology/Path Connectedness

Topology Path Connectedness Connectedness Compactness Definition A topological space is said to be path connected if for any two points there exists a continuous function such that and Example All convex sets in a vector space are connected because one could just use the segment connecting them, which is . The unit square defined by the vertices is path connected. Given two points the points are connected by the function for .The preceding example works in any convex space (it is in fact almost the definition of a convex space). Adjoining Paths Let be a topological space and let . Consider two continuous functions such that , and . Then the function defined by Is a continuous path from to . Thus, a path from to and a path from to can be adjoined together to form a path from to . Relation to Connectedness Each path connected space is also connected. This can be seen as follows: Assume that is not connected. Then is the disjoint union of two open sets and . Let and . Then there is a path from to , i.e., is a continuous function with and . But then and are disjoint open sets in , covering the unit interval. This contradicts the fact that the unit interval is connected. Exercises Prove that the set , where is connected but not path connected. Topology Path Connectedness Connectedness Compactness Topology