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708 Review Sheet for MA 57100 at Purdue

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This 7 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at Purdue University taught by a professor in Fall. Since its upload, it has received 29 views.

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Date Created: 02/06/15
Review problei ns for Midterm 2 Unless otherwise stated you may use anything in Munkres s book but be care ful to make it clear what fact you are using When you use a set theoretic fact that isn t obvious be careful to give a clear explanation O1 l Compactness Let X be the closed unit disk in R2 and define an euivalence relation on X bv 4 a w I if either a I or a and I are both on the unit circle Let X be the partition of X into the equivalence classes of N Show that X With its quotient topology is hoi neoi norphic to the one point compactification of the open unit disk Prove that every coi npact subset of a Hausdorff space is closed Show that if Y is compact then the projection map X X Y gt X is a closed map Let X be a locally compact space Explain how to construct the one point compact ification of X and prove that the space you construct is really compact you do not have to prove anything else for this problem 00 Show that if HXT is locally compact and each X7 is nonei npty then each X7 is nl locally compact and X7 is compact for all but finitely many n Let X be a Hausdorff space and let and B be disjoint compact subsets of X Prove that there are open sets U and V such that U and V are disjoint C U and B C Let X be a coi npact metric space and let U be a covering of X by open sets Prove that there is an 6 gt 0 such that for each set S C X With diameter lt 6 there is a U 6 U With S C U This fact is known the Lebesgue nui nber lei ni naquot Let X be a locally coi npact Hausdorff space let Y be any space and let the function space CX Y have the coi npact open topology Prove that the map a X X CXY gt Y defined by the equation xvfl flat is continuous Let X be a coi npact Hausdorff space Let f X gt Y be a continuous Closed surjection Prove that Y is Hausdorff 39N 3 2 x M 1mm 0x Sugugmum X 30 M 333 uado 1w 8 mag 11an g x 02 Sugugmum A x X 30 ms uado 1w aq N mi pmz X 9 02 mi 3312d1mm 3 mm samds pmg ogxio aq 3 pm X mi mnmaq aqnl mp aAmd 71 39 1 3 g mm 039 UP sg amp 3mg 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mm 11 pm a1hmzxaw3unm p ng xo aAmd 39 81 017 mm lt Ox X 11 f uaq 1 81 f 11 11am mmxmms Sumoutg mp SI 39027 03 02 mg ngm Hanging snonugum p aq A lt X f 331 pm A 9 017 X 9 0x 331 samds pag qodm aq A pm X 331 39 3906 3961 3981 3991 and U V U D V are path connected Let i1 U D V gt U 172 U D V gt V jl U gt X and 2 V gt X be the inclusion maps Suppose that Mt mU Vim gt W1Ux0 is onto Prove using the Seifei t van Kampen theoi ei n that 726 W1Vx0 gt W1Xx0 is onto

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