Exercise 24, § 4

Let $X$ be a locally compact and locally connected space, $Y$ a separated space, $f : X \to Y$ a continuous mapping. Let $x$ be a point of $X$ such that the subspace $f^{-1}(f(x))$ of $X$ is totally disconnected. Show that when $U$ runs through a fundamental system of neighbourhoods of $f(x)$ in $Y$, the connected components of $x$ in the sets $f^{-1}(U)$ form a fundamental system of neighbourhoods of $x$. (Reason by contradiction, using Exercise 14 of II, p. 38).