Example that is neither an Fσ nor a Gδ but it is an Fσδ.

Question

I can not find any example which can satisfy the given situation.

If there is any example then please explain that to me.

A subset of R that is neither an Fσ nor a Gδ but it is an Fσδ.

Also if anybody can explain diagram of fig 1.1.enter image description here

A simple counterexample is provided here – Alex Ravsky Oct 18 '17 at 06:37 — Alex Ravsky, Oct 18 '17 at 06:37
But also prove that this example belongs to Fσδ. – Shrawan Oct 18 '17 at 06:44 — Shrawan, Oct 18 '17 at 06:44

score 1 · Answer 1 · answered Oct 18 '17 at 06:37

For a trivial and uninteresting example, let $A\subset(0,1)$ be an $F_\sigma$ but not a $G_\delta,$ and let $B\subset(2,3)$ be a $G_\delta$ but not an $F_\sigma;$ then $S=A\cup B$ will be an $F_{\sigma\delta}$ and a $G_{\delta\sigma}$ but neither a $G_\delta$ nor an $F_\sigma.$ For instance, $S=((0,1)\cap\mathbb Q)\cup((2,3)\setminus\mathbb Q).$

Example that is neither an Fσ nor a Gδ but it is an Fσδ.

1 Answers1