Infinite Hausdorff Spaces: Mutually Disjoint Open Subsets

[mathboy]

Prove that an infinite Hausdorff space has an infinite collection of mutually disjoint open subsets.

 

[mathboy]

Each of the N disjoint open sets is itself an infinite Hausdorff space, so I can break each open set into smaller disjoint open sets, right?


Assume there are N disjoint open subsets {U1,...,Un} in X. Let x,y be in U1. Since X is Hausdorff there exists disjoint open subsets A and B containing x and y, respectively. Then (U1 intersect A) and (U1 intersect B) can replace U1 in the collection, giving us N+1 disjoint open subsets. Thus there is no maximum number of disjoint open sets in X.