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.

Consider a space like the closed topologist's sine curve, i.e., the closure $\mbox{cl}(G)$ where $G$ is the graph of $\sin(1/x)$ for $0 < x \le \pi$. This space is connected, so every point has a connected neighbourhood, namely the whole space, but has points (in this case the points that lie on the $y$-axis) for which all sufficiently small neighbourhoods are disconnected.
