I am working in some cardinal functions in topological spaces, and I need to either define a new cardinal function or use an existing one that behaves in the following way:
Lets say $\varphi$ is my desired cardinal function, and let $X$ be a topological space with $\varphi(X)=\kappa$ for some cardinal $\kappa$, I need that for every family of open sets $\mathcal{F}$ such that $|\mathcal{F}|\leq \kappa$ it holds that $$\bigcap_{U\in\mathcal{F}}U $$ is an open set.
I have been looking around and it seems like there is nothing that behaves this way, and yet I cannot come up with some cardinal function that makes this statement true; any help is appreciated, Thanks in advance.
I suggest the definition be g(X) =
max{ $\kappa$ : for all F subset topology X, (|F| < $\kappa$ implies $\cap$F is open) }. Thus for all spaces S with an infinite topology, $\aleph_0$ <= g(S).
If S is an Alexdranov space (all intersections of open sets are open), g(S) is the successor cardinal of |topology S|.
Is this cardinal called the intersectuallity of the space?
Are there examples of spaces with uncountable intersectuallity other than Alexandrov spaces?
What use is there for this function?
Does intersectuallity, as some cardinal functions may do, disregard finite values by making them $\aleph_0?$