Question

Exercise 1.9.52. Give an example of a path connected space which is not locally path connected

Answer 1

A topological space X is said to be locally connected at a point x∈X if, for every neighborhood U of x (i.e. open set U such that x∈U), there exists a connected neighborhood V of x such that V⊂U. If X is locally connected at each of its points, then it is said to be locally connected.

An example of a path connected space that is not locally path connected is the comb space: if K = {1/n |n is a natural number}, then the comb space is defined by:

C = (K × [0,1]) ∪ ({0} × [0,1]) ∪ ([0,1] × {0}]

The comb space is path connected (this is trivial) but locally path connected at no point in the set A = {0} × (0,1]. It is however locally path connected at every other point.