Making up my math foundation...
Why do we require "finite" intersections when specifying the open sets in topology?
> One major reason: it makes topological spaces into a generalization of metric spaces. Indeed, in a metric space, arbitrary unions and finite intersections of open sets are open, but countable intersections of open sets often aren't open. For example, in the real line with the usual Archimedean metric, the intersection of (0, 1 + 1/n) over all positive integers n is (0, 1], which isn't open.
Why do we require "finite" intersections when specifying the open sets in topology?
> One major reason: it makes topological spaces into a generalization of metric spaces. Indeed, in a metric space, arbitrary unions and finite intersections of open sets are open, but countable intersections of open sets often aren't open. For example, in the real line with the usual Archimedean metric, the intersection of (0, 1 + 1/n) over all positive integers n is (0, 1], which isn't open.