TITLE: In a topological space, why the intersection only has to be finite?
QUESTION [7 upvotes]: I was wondering if there is a reason why in the definition of a topologic space, the reunion of any collection of open sets is also an open set but the intersection has to be finite to be a set.
Is there because of the definition of an open set itself ? I have always seen it as a definition, but now I want to know why.
Thanks for your help.

REPLY [2 votes]: This is an alternative approach to the definition of topology, thinking of topology in general as representing intuitionist logic. This by no means should be your primary argument, I just always liked this notion.
You can think of open sets as sets for which we can have "finite witnesses." That is, we can show a point $x$ is in the open set $U$ with a finite amount of information. 
Then if $\mathcal U$ is a set of open subsets of a topological space, to show that $x\in\bigcup \mathcal U,$ we need only finite information that $U\in\mathcal U$ and finite information showing that $x\in U.$
On the other hand, if $\mathcal U$ has infinitely many sets, to show that $x\in\bigcap \mathcal U,$ we'd have to have an infinite list of finite information showing $x\in U$ for each $U\in\mathcal U.$ So there is no reason to think we can provide for such a general $x$ a finite proof that $x\in\bigcap \mathcal U.$