Question: What is the difference between talking about "Any collection $\{G_\alpha\}$ of open sets" and "Any finite collection of $G_1,\dots,G_n$ of open sets"?
I imagine they are highlighting the difference betweem finite and infinite collections. But I see no such point, since the first could be finite.
Is there reason for this wording?
This is from Rudin - PMA. Theorem 2.24:
For any collection $\{G_\alpha\}$ of open sets, $\cup_\alpha G_\alpha$ is open.
For any collection $\{F_\alpha\}$ of closed sets, $\cap_\alpha F_\alpha$ is closed.
For any finite collection $G_1,\dots,G_n$ of open sets, $\cap_{i=1}^n G_i$ is open.
For any finite collection $F_1,\dots,F_n$ of closed sets, $\cap_{i=1}^n F_i$ is closed
The first could indeed be finite, but this wording emphasises that the collection need not be finite - whereas when the wording "for any finite collection" is used, this is a necessary condition for the conclusions of the theorem to hold.