Topology/Path Connectedness

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Topology
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Definition[edit]

A topological space is said to be path connected if for any two points there exists a continuous function such that and

Example[edit]

  1. All convex sets in a vector space are connected because one could just use the segment connecting them, which is .
  2. 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[edit]

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[edit]

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[edit]

  1. Prove that the set , where
    is connected but not path connected.


Topology
 ← Connectedness Path Connectedness Compactness →