let $A,B$ be subsets of topological space $X$. Does it follow
$\partial (A \cap B) = \partial A \cap \partial B $ ?
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$\begingroup$ Try to find counterexamples for both inclusions. $\endgroup$– Stefan HamckeOct 21, 2013 at 18:45
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3 Answers
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On these types of problems it's always best to draw picture when there's an opportunity to!
answered Oct 21, 2013 at 19:36
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$\begingroup$ @user153330 yes, ms paint, I would use Paint 3D that comes with windows nowadays. $\endgroup$ Oct 29, 2018 at 21:44
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Let $X = \mathbb{R}$, $A = \mathbb{Q}$, and $B = \mathbb{R} \setminus \mathbb{Q}$ be the rationals and irrationals. What do you get?
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Consider $X=\mathbb{R}$, with the usual topology. If $A=[-1,1]$ and $B=[-2,2]$, then $\partial(A\cap B)=\partial([-1,1])=\{-1,1\}$; $\partial A=\{-1,1\}$; and $\partial B=\{-2,2\}$.
But certainly, $\{-1,1\}\neq \{-1,1\}\cap\{-2,2\}$.
answered Oct 21, 2013 at 18:43