On the definition of locally connectedness. Why don’t we just define the locally connectedness the same way we define locally compactness, that is, every point has a connected neighborhood?
On the wiki page the weak locally connectedness and connectedness are proved to be “almost identical”, but it does not mention the motivation of the definition.

I think the crucial fact is that (reference)

A space is locally connected if and only if for every open set U, the connected components of U (in the subspace topology) are open.

With the other definition (the weak one), this would be false.
