# A $G_\delta$ and meager set in $\mathbb{R}$

I've thought about it, and I'm not sure such sets exist. I mean, obviously the finite sets satisfy both definitions, buy I'm searching a non-trivial example.

The definitions are: A set $$M \subseteq \mathbb{R}$$ its a meager set if exists a sequence $$\{F_n\}$$ of nowhere dense (i.e., with empty interior) with $$M = \bigcup F_n$$; and a set $$X \subseteq \mathbb{R}$$ its a $$G_\delta$$ set if exists a sequence $$\{A_n\}$$ of open sets with $$M = \bigcap A_n$$.

If any have an idea or some reference, I would appreciate the information. Thanks!

• Right you are... Sep 10 '20 at 16:15
• FYI, a $G_{\delta}$ meager set is nowhere dense. The converse doesn't hold, but not because of size considerations (because every nowhere dense set is contained in a closed nowhere dense set, and a closed nowhere dense set is certainly a $G_{\delta}$ meager set), but because of "descriptive set complexity" considerations (there exist nowhere dense sets that are not $G_{\delta},$ indeed not even Borel). Sep 10 '20 at 18:21

A more interesting example is the Cantor set, it is a nowhere dense (hence meager) subset of the reals with the same cardinality as the whole of $$\Bbb R$$, and it also a $$G_\delta$$ set since every closed set is $$G_\delta$$ in a metric space. By considering a fat Cantor set instead of the standard middle-thirds construction you can even get examples with positive measure.
Note that in general, by the Baire category theorem, it is much easier to produce comeager $$G_\delta$$ sets (hence meager $$F_\sigma$$ sets)