Exercise 3, § 1

If $A$ is any subset of a topological space $X$, let $\alpha(A)$ denote $\overline{\dot{A}}$ and $\beta(A)$ denote $\dot{A}$. Then $A \subset B$ implies $\alpha(A) \subset \alpha(B)$ and $\beta(A) \subset \beta(B)$.

a) Show that if $A$ is open then $A \subset \alpha(A)$, and that if $A$ is closed then $\beta(A) \subset A$.

b) Deduce from a) that if $A$ is any subset of $X$, then $\alpha(\alpha(A)) = \alpha(A)$ and $\beta(\beta(A)) = \beta(A)$.

c) Give an example of a set $A$ (on the real line) such that the seven sets $A, \mathring{A}, \overline{A}, \alpha(A), \beta(A), \beta(\overline{A}), \alpha(\mathring{A})$ are all distinct and satisfy no relations of inclusion except the following: $\mathring{A} \subset A \subset \overline{A}, \mathring{A} \subset \alpha(\mathring{A}) \subset \beta(A) \subset \overline{A}, \mathring{A} \subset \alpha(A) \subset \beta(\overline{A}) \subset \overline{A}, \alpha(\mathring{A}) \subset \alpha(A), \beta(A) \subset \beta(\overline{A})$.

d) Show that if $U, V$ are two open sets such that $U \cap V = \varnothing$, then also $\alpha(U) \cap \alpha(V) = \varnothing$ [use b)].