A variant of Baire Category Theorem states that if a complete metric space is a countable union of closed sets then at least one of the sets has a non-empty interior.
In my study I have a Banach space which is a countable union of closed sets $F_n$, with the property that for every $x \in F_n$, if $\lambda \in \Bbb{K}$ with $| \lambda| \geq 1$ then $\lambda x \in F_n$. I would need to find a set out of these which contains a circle(not necessarily a disk) around the origin(not necessarily centered in the origin), but containing the origin in its interior. I know this is almost impossible using only these conditions, but maybe I can find more properties of the sets $F_n$ before leaving this lead.
My question is like this:
Are there any theorems similar to Baire Category theorem which can prove something like this? Even if there is no theorem that directly solves the problem above, I am interested in any exotic variant of Baire Category theorem, in which maybe the sets $F_n$ have additional properties.
References are welcome. Thank you.