Subset of a $F_{\sigma}$ set is $F_{\sigma}$

Question

Suppose that $X$ is a metric space. Is it always true that for any $F_{\sigma}$ set $A$, any subset $B \subset A$, $B$ is $F_{\sigma}$? It seems correct to me but I have no idea how to prove it.

$X$ itself is $F_{\sigma}$! So you are asking us if every subset is $F_{\sigma}$, which is not true in general. — Crostul, Jul 14 '15 at 10:04

score 3 · Accepted Answer · edited Apr 13 '17 at 12:21

3

I don't think so. Observe that $\mathbb R$ is $F_\sigma$, but the subset $\mathbb R \setminus \mathbb Q$ is not.

edited Apr 13 '17 at 12:21

Community

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answered Jul 14 '15 at 09:24

Potato

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Subset of a $F_{\sigma}$ set is $F_{\sigma}$

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