First of all, rest assured that I have used the search tool and read Why do we require a topological space to be closed under finite intersection? and https://mathoverflow.net/questions/19152/why-is-a-topology-made-up-of-open-sets.
I'm looking for a more logical approach on this issue. I understand intuitively the reasons why an intersection of infinitely many open sets may fail to be open, but only in the case of a topology induced by a metric. In the general sense, what would be a somewhat rigorous argument? I feel that some counterexamples aren't enough for me to deeply understand the motives. I've read somewhere that the intersection of an infinite family of sets is related with an infinite conjunction, which is not allowed in the usual logical framework we use for mathematics. Is this the issue? Could you explain it in more detail? Are infinite disjunctions permissible, then? Finally, if we were to work within an infinitary logic, would this "issue" disappear, and consequently, would the building of topology become somewhat trivial, or less rich and interesting?