Given a set $X$ we say that a collection $\cal C$ in $X$ is totally closed with respect union or intersection if for any subcollection $\cal S$ of $\cal C$ their union or intersection is in $\cal C$ and so we immediately observe that $\mathcal P(X)$ is trivially totally closed with respect union and intersection so that the collections $\mathfrak C_\cup(X)$ and $\mathfrak C_\cap(X)$ of totally closed collection in $X$ with respect union or intersection are not empty and not disjoint too. Anyway, let's we prove the following result.
Proposition
For any set $X$ the collection $\mathfrak C_\cup(X)$ is totally closed with respect union whereas the collection $\mathfrak C_\cap (X)$ is totally closed with respect intersection.
Proof. So let be $\frak C_\cup$ is a subset of $\mathfrak C_\cup(X)$ so that we observe that if $\cal U$ is a subset of $\bigcap\frak C_\cup$ then for any $\cal C$ in $\frak C_\cup$ the inclusion $$ \cal U\subseteq C $$ holds and thus (remember that $\cal C$ is in $\frak C_\cup$ and so is totally closed with resepct union) $\bigcup\cal U$ is in $\cal C$ and so in $\bigcap\frak C_\cup$ and this proves that $\bigcap\frak C_\cup$ is totally closed with respect union. Analogously it is possibile to prove that if $\frak C_\cup$ is a subset of $\mathfrak C_\cup(X)$ then even $\bigcap\mathfrak C_\cup$ is there contained.
So I ask if I well proved the last proposition and thus I ask to prove or disprove (I am sure it is false) if the collections $\mathfrak C_\cup(X)$ and $\mathfrak C_\cap(X)$ are not totally closed with respect union: could someone help me, please?
Trivially any topology is totally closed with respect union but unfortunately union of topologies is not a topology and in particular this simple counterexample shows that the union of two topologies is not (totally) closed with respect union so that generally an union of collections which are totally closed with respect union is not totally closed with respect union; moreover, if $\cal C$ is a collection on a set $X$ totally closed with respect union then the complement collection $\mathcal C^C$ is totally closed with resepect intersection and vice versa so that by the linked counterexample we argue that union of collection which are totally closed with respect intersection is not totally closed with respect intersection.