Consider tw0 sets $A, B$ from the topological space X. Is the following true: $$ \mathbb{cl} (A \setminus B) = \mathbb{cl}\, A \setminus \mathbb{cl} \,B $$
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1Take $\mathbb{R}$ with the standard topology, let $A$ be a closed and $B$ a single point which belongs to $A$. – MSDG May 11 '19 at 19:41
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Let $A$ be the rationals, $B$ the irrationals in $\Bbb R$ usual topology.
$A \setminus B = \Bbb Q$ with closure $\Bbb R$
however, both the closure of $A$ and $B$ are $\Bbb R$ so the right hand side is empty.
Henno Brandsma
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Thanks for the answer. Are you aware of a formula? I couldn't find so far. – user48672 May 11 '19 at 20:47
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@user48672 there is none AFAIK. One inclusion does always hold. – Henno Brandsma May 12 '19 at 05:52