Q:

A question based on Interior , Closure of sets

This particular question is part of my real analysis assignment which couldnot be discussed .
Edit: In all four options It would be distinct instead of disjoint. Sorry for typo!!

For a set A which is proper subset of $\mathbb{R}$ , denote by Cl(A) the closure of A , and by Int(A) the interior of A . There is a set A ( proper subset of $\mathbb{R}$) such that

A, cl(A) and Int(A) are pairwise distinct.
A, cl(A) and Int(A)  and Cl(Int(A)) are pairwise distinct.
A, cl(A) and Int(A) and Int(Cl(A)) are pairwise distinct.
A, cl(A) and Int(A), Cl(Int(A)), Int(Cl(A)) are pairwise distinct.

For 1. taking $\mathbb{Q}$ was sufficient . But along with this example , I tried some more examples which could not satisfy 2.,3.,4. .
While checking answers I got 2.,3.4. false ie they can never be satisfied .
Can anyone please tell what is reasoning behind it

A:

Look at Kuratowski's $14$-set theorem (e.g. here)
where you can also find the example
$$A = (0,1) \cup (1,2) \cup \{3\} \cup ([4,5]\cap \Bbb Q)$$
as a set that works for all your questions (of course "disjoint" you wrote is nonsense, and should be "distinct").

