The differences between connectedness and path-connectedness, between connectedness and local connectedness, and between local connectedness at a point and weak local connectedness at a point are illustrated by examples, and it is shown that a metric space is locally connected at every point if and only if it is weakly locally connected at every point. It is also shown that if in addition such a metric space is connected and locally compact, it is path-connected. This result is key for all that follows.