The intersection of an empty collection of subsets of X is equal to X?

[ModernLogic]

Hi,

I'm reading HL Royden's real analysis, though my question pertains more to set theory.

Let X be a set. Then the intersection of an empty collection of subsets of X is equal to X. I understand this is not an intersection of empty subsets but it is still very counter-intuitive. Can anyone provide insight?

 

[Hurkyl]

Well, what element wouldn't be in the intersection?

 

[g_edgar]

Consider an intersection of lots of sets. If you take fewer sets, the intersection is bigger, right? So, in the ultimate case, when you take the fewest sets of all, the intersection is as big as possible.