Problem 4. Let $X = \mathbb{R}\setminus\{0\}$, $Y = \mathbb{R}$, and consider $f: X \to Y$ with $f(x) = 1/x$. We use the standard topology on $\mathbb{R}$ (and on X, the associated subspace topology), so we know that $f$ is continuous. Now let $A = \mathbb{N} = \{1, 2, 3, ...\} \subset X$, and $V = f(A) \subset Y$. Then according to Thm 3.13, we must have $\overline{f(A)} \subset f(\overline{A})$ and $\overline{f^{-1}(V)} \subset f^{-1}(\overline{V})$. Show that in the present case, one of these is in fact an equality, but the other is not.
