Exercise 1, § 2

Let $f$ be a mapping of a topological space $X$ into a topological space $X'$. Prove that the following statements are equivalent: a) $f$ is continuous on $X$. b) For every subset $A'$ of $X'$, $\overline{f^{-1}(A')} \subset (\overline{f^{-1}(A')})^0$. c) For every subset $A'$ of $X'$, $\overline{f^{-1}(A')} \subset \overline{f^{-1}(\overline{A'})}$. Show by an example that, if $f$ is continuous, the sets $\overline{f^{-1}(A')}$ and $\overline{f^{-1}(\overline{A'})}$ can be distinct.