does the closure of interior of a set equal to closure of this set?

Question

Does the $\text{Cl}(\text{Int} A)=\text{Cl}(A)$? Here "Cl" denotes closure, "Int" denotes interior.

That is not a duplicate of the question of "does the closure of interior of a set equal t the interior of closure of this set".

I have a problem when trying to understand the Sanov's theorem in https://blogs.princeton.edu/sas/2013/10/10/lecture-3-sanovs-theorem/

about the Sanov's theorem. $-\underset{Q\in int \Gamma}{\inf} D(Q||P)\leq \underset{n\rightarrow\infty}{\lim\inf}\frac{1}{n}\log P\leq -\underset{Q\in cl \Gamma}{\inf}D(Q||P)$

inf over $\Gamma$ equlas to inf over $cl \Gamma$, should the left hand side and right hand side be equal?

Thank you！

The answer to the question in the title is generally no. For example, a closed set could have empty interior. — Matt Samuel, May 24 '15 at 04:14
possible duplicate of Does the 'closure of the interior' equal the 'interior of the closure'? — , May 24 '15 at 05:15
@avid19 This one doesn't have the "interior of the closure" part. — , May 24 '15 at 05:39
@avid19: No, it is obviously not a duplicate of that question. — Brian M. Scott, May 24 '15 at 05:40

score 2 · Accepted Answer · answered May 24 '15 at 05:12

2

The answer is no. Look at example of $A=\mathbb{Q}$.

$\overline{A^o}=\emptyset$ but $\overline{A}=\mathbb{R}$, where $\overline{A}$ is closure of $A$ and $A^o$ is interior of $A$.

answered May 24 '15 at 05:12

Thank you！ This is a good example! – Shaofeng Zou May 25 '15 at 14:42

does the closure of interior of a set equal to closure of this set?

1 Answers1