Vito Prosciutto: Teaching community college math on the road to a PhD.

Friday, June 03, 2005

Definition of a topology: A question 

I've been puzzling over the definition of a topology from Munkres (section 12, p. 76):
A topology on a set X is a collection T of subsets of X having the following properties:
  1. The empty set and X are in T.
  2. The union of the elements of any subcollection of T is in T.
  3. The intersection of the elements of any finite subcollection of T is in T.
It's that third item. Why must it be a finite subcollection? Can anyone give me a straightforward counterexample where the intersection of elements of an infinite subcollection doesn't work?

I suppose my other problem, is that Munkres hasn't really explained what a topology is for yet. I kind of feel like I'm looking at the abstract definition of a geometry without ever having seen any geometry before.

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