A subset $C \subseteq X$, a complete metric space (particularly, a reflexive Banach space), is generic if it contains a dense $G_\delta$ subset. One advantage of generic sets is that they are a family of dense sets that are closed under countable intersections (as opposed to dense sets, which aren't even closed under finite intersection).
I'm looking for a local equivalent around a point, that is, sets $C$ would possess this property at points $x \in C$. I would want the following facts to hold:
- If $C$ has this property at $x$, then $x$ is an accumulation point of $C$, and
- If $(C_n)_{n\in \mathbb{N}}$ is a countable family of sets that possess this property at $x$, then the same goes for $\bigcap_{n \in \mathbb{N}} C_n$. (I would settle for finite unions instead of countable unions too.)
Clearly genericness on some fixed radius ball would satisfy both of these conditions, but I'm wondering if there's something weaker. In particular, it'd be nice if there didn't have to be any fixed accumulation points other than $x$, for example (accumulation points other than $x$ are fine, but it shouldn't be a requirement for any fixed point other than $x$ to be an accumulation point).