Locally Connected Space - More On Local Connectedness Versus Weak Local Connectedness

More On Local Connectedness Versus Weak Local Connectedness

Theorem

Let X be a weakly locally connected space. Then X is locally connected.

Proof

It is sufficient to show that the components of open sets is open. Let U be open in X and let C be a component of U. Let x be an element of C. Then x is an element of U so that there is a connected subspace A of X contained in U and containing a neighbourhood V of x. Since A is connected and A contains x, A must be a subset of C (the component containing x). Therefore, the neighbourhood V of x is a subset of C. Since x was arbitrary, we have shown that each x in C has a neighbourhood V contained in C. This shows that C is open relative to U. Therefore, X is locally connected.

A certain infinite union of decreasing broom spaces is an example of a space which is weakly locally connected at a particular point, but not locally connected at that point.