In my research I was faced with the problem (as a special example and a pattern for more general problems) whether the family $\operatorname{GR} ( \Delta \times^{\mathsf{FCD}} \Delta)$ of sets is closed regarding finite intersections.
To make the problem accessible for these who have not read my book, I will define it in elementary terms:
$\operatorname{GR} ( \Delta \times^{\mathsf{FCD}} \Delta)$ is a set of binary relations on $\mathbb{R}$ defined by the formula:
$$f \in \operatorname{GR} ( \Delta \times^{\mathsf{FCD}} \Delta) \Leftrightarrow \forall X \in \mathscr{P} \mathbb{R} : ( (\forall D \in \Delta : X \cap D \neq \varnothing) \Rightarrow f [X] \in \Delta)$$
where $\Delta$ is the filter of neighborhoods of zero on real line, that is the filter generated by the set $\{(-\epsilon;\epsilon) \,|\, \epsilon\in\mathbb{R}, \epsilon>0 \}$ and $f[X] = \{ y \,|\, \exists x\in X: (x,y) \in f \}$.
Is $\operatorname{GR} ( \Delta \times^{\mathsf{FCD}} \Delta)$ closed under finite intersections?
It seems the following. Put $f_1=\{(x,y)\in\mathbb R^2:|x|\le |y|$ or $y=0\}$ and $f_2=\{(x,y)\in\mathbb R^2:|x|\ge |y|$ or $x=0\}$. Then $f_1, f_2\in\operatorname{GR} ( \Delta \times^{\mathsf{FCD}} \Delta)$ but $f_1\cap f_2=\{(x,y)\in\mathbb R^2:|x|=|y|$ or $x=0$ or $y=0\}\not\in\operatorname{GR} ( \Delta \times^{\mathsf{FCD}} \Delta)$.