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！
Best Answer
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 $A$.