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Unformatted text preview: MATH 7338 HOMEWORK #1 Due date: September 22, 2009 Work the following problems and hand in your solutions. You may work together with other people in the class, but you must each write up your solutions independently. A subset of these will be selected for grading. Write LEGIBLY on the FRONT side of the page only, and STAPLE your pages together. 1. Let E be a collection of subsets of a set X whose union is X. The topology T ( E ) generated by E is defined to be the smallest topology on X that contains E (which means that T ( E ) is the intersection of all the topologies T that contain E as a subset). Show that the topology T ( E ) generated by E is set of all unions of finite intersections of elements of E : T ( E ) = braceleftbigg uniontext i ∈ I n intersectiontext j =1 E ij : I arbitrary, n ∈ N , E ij ∈ E bracerightbigg . 2. Let X and Y be topological spaces, and set B = { U × V : open U ⊆ X, open V ⊆ Y } ....
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 Fall '09
 HEIL
 Math, Topology, Topological space, Compact space, Banach space, Topological vector space

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