Proposition 7

A locally path-connected topological space is locally connected. Its connected components coincide with its path-connected components. In particular, if a locally path-connected topological space is connected, it is path-connected.

The first assertion follows from prop. 4. Let us prove the second assertion. Let X be a locally path-connected topological space and let C be a path-connected component of X. Every point of C possesses a path-connected neighbourhood, hence contained in C; consequently, C is an open subset of X. The path-connected components forming a partition of X, such a component is also closed. Since it is connected (III, p. 258, prop. 4), it is a connected component (TG, I, p. 83). The third assertion follows from the second.

Thus we see that there is no ambiguity in saying that a topological space is connected and locally path-connected.