Give an example of a set $A$ for which the following sets are pairwise different.

Question

Give an example of a set $A$ for which the sets:

$A, \text{Int}(A), \overline A, \text{Int}(\overline A), \overline {\text{Int}(A)}, \text{Int}(\overline{\text{Int}( A)}), \overline{\text{Int} \big( \overline A\big)} $

Are pairwise different.

My prof gives a hint, the set $A$ will be $\mathbb{Q}\cap B$ where $B \subset \mathbb{R}$, but $\text{Int}(\mathbb{Q})=\phi$ and then $\text{Int}(\mathbb{Q} \cap B)=\phi$, also $\overline{\text{Int}(\mathbb{Q} \cap B)}=\phi$, what’s the wrong?

I need a hint, please.

Thanks.

Answer 1

Take $A= [0,1) \cup (1,2) \cup \{3\} \cup \left(\mathbb{Q} \cap (4,5)\right)$:

• $\overline{A} = [0,2] \cup \{3\} \cup [4,5]$

• $\operatorname{int}(\overline{A})= (0,2) \cup (4,5)$,

• $\overline{\operatorname{int}(\overline{A})}=[0,2]\cup [4,5]$

• $\operatorname{int}(A) = (0,1) \cup (1,2)$,

• $\overline{\operatorname{int}(A)}= [0,2]$,

• $\operatorname{int}(\overline{\operatorname{int}(A)})=(0,2)$.