If $ \sup(A)=\sup(B)$ and $\inf(A)=\inf(B)$ does that mean that $A=B$?

Question

If $A=B$ then its easy to show that $\sup(A)=\sup(B)$ and $\inf(A)=\inf(B)$. However, is the opposite of this statement also true?

i.e. If $\sup(A)=\sup(B)$ and $\inf(A)=\inf(B)$ then $A=B$?

Answer 1

No, and the reason is that this information about the extreme values of the sets, does not say anything about the other values of the sets that are between these extreme values (or bounds). Some examples

1. $A=[0,1]$ and $B=(0,1)$,
2. $A={-1}\cup[0,1]\cup {2}$ and $B[-1,2]$
3. $A=(0,1) \cup (2, +\infty)$ and $B=[0,+\infty)$,
4. $A=\left\{\frac1n: n \in \mathbb N\right\}$ and $B=[0,1]$,
5. $A=[0,1] \cap \mathbb Q$ and $B=[0,1]\cap \mathbb {R\setminus Q}$.
6. $\ldots$ many (many) more.

But, if $\sup A= \sup B$ and $\inf A=\inf B$ holds and you know that the sets $A, B$ have other properties in common: for example that they are both open connected intervals, then you can perhaps (in this special case) infer that they are the same.