Topology/Path Connectedness

 ← Connectedness Path Connectedness Compactness → 

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.

 ← Connectedness Path Connectedness Compactness →