I have also asked this in MSE, but it seems to me that my question wasn't very well received there and I think someone in here will be able to answer it more quickly, hence this post.
I don't understand some facts about invariant subsets of a local group action. Basically (to save you reading definitions) local actions are germs of partial actions which in turn are just like regular actions with the difference that the domain doesn't need to be all of $G\times M$. For the sake of context I'm going to reproduce the definition of a local group action on a topological manifold.
Definition. Let $G$ be a Lie group and $M$ a topological. A partial action of $G$ on $M$ is a continuous map
$\rho: \mathrm{dom}(\rho)\rightarrow M$
where $\mathrm{dom}(\rho)\subset G\times M$ is an open subset containing $\{e\}\times M$ ($e$ es the identity element of $G$). The map $\rho$ satisfies the following two conditions:
(i) $\rho(e,x)=x$ for all $x$
(ii) If $(g_2,x)\in \mathrm{dom}(\rho)$ and $(g_1,(\rho(g_2,x)) )\in\mathrm{dom}(\rho)$, then $(g_1g_2,x)\in \mathrm{dom}(\rho)$ and $\rho(g_1,(\rho(g_2,x)) )=\rho(g_1g_2,x)$.
Definition. There is an equivalence class of partial actions: Two partial actions $\rho_1$ and $\rho_2$ are equivalent if there is an open subset $D\subset \mathrm{dom}(\rho_1)\cap\mathrm{dom}(\rho_2)$ such that $\rho_1|_{D}=\rho_2|_{D}$. An equivalence class $[\rho]$ of partial actions is called a local action.
An subset $Y$ of $M$ is $[\rho]$-invariant if there is a representative $\varphi\in [\rho]$ such that whenever $y_0\in Y$ and $(g,y_0)\in \mathrm{dom}(\varphi)$ then $gy_0\in Y$.
Actual Question. Supposedly the intersection of any number of invariant subsets is invariant. This is where I am confused. For a finite collection if subsets this is very clear: If $\{Y_i\}_{i=1}^n$ are $[\rho]$-invariant then for each $i$ there is a $\varphi_i\in[\rho]$ satisfying the definition of invariance. We can take $D= \cap_{i=1}^n\mathrm{dom}(\varphi_i)$ and restrict any of the $\varphi_i$ to obtain the required partial action. IF we have an infinite number of invariant subsets then this is not so clear anymore since the intersection of the domains could be non-open (it could even be $\{e\}\times M$).
How can I prove this?
I've tried a proof by contradiction: Supposing $\cap Y_i$ is not invariant would yield a sequence $\{g_i\}$ in $G$ which (I think but I'm not sure) could be made to converge to $e$. After that if I try to use the continuity of the $\varphi_i$ I could obtain a contradiction to (i) in the definition of partial action.