I know questions similar to this have been asked here but, is it possible to find a subset of a topological space such that its closure of interior and interior of closure does not contain each other? For example if $X=\mathbb{R}$, $A=\mathbb{Q}$, the closure of interior of A would be contained in the interior of closure of A.


  • $\begingroup$ What do you mean "does not contain each other?" That neither contains the other? $\endgroup$
    – 5xum
    Commented Feb 23, 2016 at 9:49

1 Answer 1


An example inside $\Bbb{R}$ is $$A=([0,1] \cap \Bbb{Q}) \cup [2,3]$$ the interior of the closure of $A$ is $(0,1) \cup (2,3)$, while the closure of the interior is $[2,3]$: these two sets are not comparable by inclusion.


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