Let $A$ and $B$ be closed sets in $\mathbb{R}$. Is $A\setminus B$ an $F_{\sigma}$ set?
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1Please iinclude your own thoughts and ideas , Also add more information as the question is too vague in its current state. See here on how to ask a good question – The Integrator May 08 '18 at 18:04
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It is both an $F_\sigma$ set and a $G_\delta$ set. – bof May 08 '18 at 23:29
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The set difference $A-B$ is the same as $A \cap B'$ where $B'$ means the complement of $B$ and so is open. So show that an open set is also an $F_\sigma$ set, which is to say a countable union of closed sets. To do this, use that an open set is a countable union of pairwise disjoint open intervals, and each of those is the union of a countable increasing family of closed intervals.
coffeemath
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What about this? Let A and B be F_{\sigma} sets in ℝ. Is A∖B an F_{\sigma} set? – APR May 16 '18 at 10:20