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

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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 $A$.

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Shaofeng Zou
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Shaofeng Zou

Updated on May 26, 2020

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  • Shaofeng Zou
    Shaofeng Zou over 3 years

    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!

    • Matt Samuel
      Matt Samuel over 8 years
      The answer to the question in the title is generally no. For example, a closed set could have empty interior.
    • Admin
      Admin over 8 years
    • Admin
      Admin over 8 years
      @avid19 This one doesn't have the "interior of the closure" part.
    • Brian M. Scott
      Brian M. Scott over 8 years
      @avid19: No, it is obviously not a duplicate of that question.
    • Shaofeng Zou
      Shaofeng Zou over 8 years
      @MattSamuel thank you.
  • Shaofeng Zou
    Shaofeng Zou over 8 years
    Thank you! This is a good example!