MAT-4710 Final: MATH 4710 App State Fall2014 Final Exam answer key
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Final exam: (15 points) give x the nite complement topology: t = {a x | x a is nite} { }. (a) show that t is a topology for x. Also, x x = is nite (it has zero elements) so x t . Then x u and x v are nite. Notice that x (u v ) = (x u ) (x v ) (by demorgan"s law) is nite since the union of two nite sets is still nite. Thus t is closed under ( nite) intersection. each x uj is nite, so the intersection of all x uj"s (which is contained in each one of them individually) must be. Finally, suppose that uj t for all j j. Again by demorgan"s (other) law, we have x [j j. This means that t is closed under arbitrary union. Let o be an open cover of x.
