Exercise 5, § 1

If $A$ is any subset of a topological space $X$, the frontier of $A$ is denoted by $\mathrm{Fr}(A)$.

a) Show that $\mathrm{Fr}(\overline{A}) \subset \mathrm{Fr}(A), \mathrm{Fr}(\mathring{A}) \subset \mathrm{Fr}(A)$, and give an example on the real line where these three sets are distinct.

b) Let $A, B$ be two subsets of $X$; show that

and give an example on the real line where these sets are distinct. If $\overline{A} \cap \overline{B} = \varnothing$, show that $\mathrm{Fr}(A \cup B) = \mathrm{Fr}(A) \cup \mathrm{Fr}(B)$.

c) If $A$ and $B$ are open in $X$, show that

and give an example on the real line where these three sets are distinct.