
 Baire Category Theorem 
  to review, a space is hausdorff if two points can be separated in disjoint open sets, and a space is locally compact if each point is contained in an open set whose closure is compact. 
  The common metric spaces are hausdorff and locally compact, but others are not, so this is not a generalization of the above. 
  The interval [0,1] is compact, hausdorff, a complete metric space, the nicest space you could think of, and the complement of any point is a dense open set, yet the intersection of these sets is empty. 
 www.mathreference.com /topms,bct.html (621 words) 
