Infinite Hausdorff Spaces: Mutually Disjoint Open Subsets

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SUMMARY

An infinite Hausdorff space contains an infinite collection of mutually disjoint open subsets. The proof begins by assuming a finite collection of disjoint open subsets {U1,...,Un} within the space X. Utilizing the Hausdorff property, for any two points x and y in U1, disjoint open sets A and B can be found, allowing for the replacement of U1 with (U1 intersect A) and (U1 intersect B), thereby increasing the number of disjoint open subsets. This process demonstrates that there is no upper limit to the number of disjoint open sets in an infinite Hausdorff space.

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