Connectedness and locally path-connectedness

Question

I'm studying algebraic topology and I have this doubt:

$X$ is a topological space connected and locally path-connected; does this imply that $X$ is path-connected ? Why ?

Meaning of locally path-connected: every point $p \in X $ has a path-connected neighbourhood.

Answer 1

Hint: Fix $a \in X$ and consider the set $$U = \{ x \in X | \mbox{ there exists a path connecting $a$ and $x$ } \}$$

Try to show that $U$ is both closed and open.