This question is inspired by the Wikipedia article on the Zariski topology: https://en.wikipedia.org/wiki/Zariski_topology
Since I know next to nothing about algebraic geometry, and no advanced abstract algebra, only the typical beginning groups, rings, fields, Galois theory sequence, any answer which "explains it like I am five" would be most appreciated. Again, please do not assume that I am intelligent in any response, it would be most helpful.
Why is the intersection of algebraic subsets of an algebraic variety again an algebraic subset?
I assume that this is true because the Wikipedia article on the Zariski topology states that the closed sets in the Zariski topology are all algebraic subsets of the algebraic variety, and obviously the arbitrary intersection of closed sets is closed in any topological space.
I want to understand this because this seems like the only answer I could find for my previous question: What is the motivation behind the arbitrary union topological axiom? Namely since the arbitrary intersection of closed sets being closed is equivalent to the arbitrary union of open sets being open, and the Zariski topology is apparently neither separable nor first/second-countable.