Exercise 2, § 3

Let $A$ and $B$ be any two subsets of a topological space $X$.

a) Show that the intersection of $A$ and the interior of $B$ with respect to $X$ is contained in the interior of $B \cap A$ with respect to $A$. Give an example where these two sets are distinct.

b) Show that the intersection of $A$ and the closure of $B$ with respect to $X$ contains the closure of $B \cap A$ with respect to $A$. Give an example where the two sets are distinct.