Let $A,B\subseteq\mathbb R^d$, $A$ closed and $B$ open. If $x\in A\cap B\neq\emptyset$ does there exist $\varepsilon>0$ such that $B_\varepsilon(x)\subseteq A\cap B$?
intesection of a closed set with an open set
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geometry
elementary-set-theory
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5If $d=1$, $A=\{0\}\subset (-1,1)=:B$, there isn't any open ball centered at $0$ and contained in $A$. – 2012-11-06
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0ah yes. good example – 2012-11-06