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Re: Nonempty intersection
Posted:
Jul 21, 2011 11:32 PM


On Thu, 21 Jul 2011, Kaba wrote: > Rotwang wrote: >> On 21/07/2011 17:37, Kaba wrote: >>> >>> Here's an exercise from the book "Introduction to Topological >>> Manifolds", page 97: >>> >>> "Let X be a compact space, and suppose {F_n} is a countable collection >>> of nonempty closed subsets X that are nested, which means that >>> F_n superset F_{n + 1} for each n. Show that intersection {F_n} is >>> nonempty." >>> >>> Hints? >> >> Suppose that the intersection is empty. Note that X is the complement of >> the empty set, and apply De Morgan's law to conclude that the union of >> the complements of the F_n's is X. Since X is compact... > > ... there is a finite set of F_n's whose complements cover X. Using De > Morgan again, the intersection of those F_n's is empty. Because the > F_n's are nested, some of the F_n's must be empty, which is a > contradiction. Therefore, the countable intersection is nonempty. > Generalization. If S is a compact space, C a collection of closed subsets, U an open subset of S and the big intersection of C a subset of U, then there's some K1,.. K_n in C with K1 /\../\ K_n subset U.



