Let $f: X \rightarrow Y$ and $g: Y \rightarrow X$ be continuous maps of topological spaces with $f \circ g=\mathrm{id}_{Y}$.

(1) Suppose that (i) $Y$ is path-connected, and (ii) for every $y \in Y$, its inverse image $f^{-1}(y)$ is path-connected. Prove that $X$ is path-connected.

(2) Prove the same statement when "path-connected" is everywhere replaced by "connected".