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Unformatted text preview: January 10, 2007 NAME:
Topology Qualifying Examination Every answer must be supported with reasoning (proof l). Please start each of the five
problems on a new page in the blue book. 1. (20 points) Let X be the set of nx n real matrices with the Euclidean topology induced from R"2 via the bijection A = (au)l—) (an,a,2,...,a2,,...,ann) , and let Y be the subspace of all matrices A satisfying ATA = 1, AT = transpose(A), I = identity matrix. a) Is Y compact?
b) Is Y normal? 0) If 5"" is the unit sphere in R" , is the functionf 2 5"" xY —> 8'” ,f(x, A) = A(x),
continuous? 2. (20 points) a) Define a locally compact space. b) Show that the rationals are not locally compact.
c) Assume as known that a finite Cartesian product of compact spaces is itself compact.
Let K. be a compact subspace of the space X, i = 1,2,...,n. Show that the subspace which isthe product of the K,~ in the product of the X, is a compact subspace.
d) If each X, is a locally compact space i = 1,2,...,n, show that the product of the X. is also locally compact. 3. (20 points) a) Define and compare the product and box topologies on R w, the countable product of real lines.
b) Let X be the subset of all sequences (x],x2,...) with x1. =2 0 for only finitely many i. Describe the closure of X in each topology. 4. (20 points) Let p : E —> B be a covering map. If U is evenly covered and {Va } is a
partition of p‘lU into slices and C is a closed set of B such that C C U , then show
(p‘lC) ﬂ Va is a closed subset of E. 5. (20 points) a) Define a covering space and a universal covering space.
b) Construct a universal covering space p : E —> B, where B = 52 U D is the union of
the unit sphere 52 in 3space and D the diameter of $2 with endpoints (—1, 0, 0) and (I, 0, 0). You may describe E as a labeled picture and describe p on each part of E.
c) Show that the two paths a (s) = (2s—], 0, 0) and [3 (s) = (cos(s+1):r, sin(s+1)n:, 0) in B are not path homotopic. d) Let X = {xER3 lllx“ s 1} be the unit ball in 3—space and let C be the circle centered at the origin of radius % in the plane x1 = 0. Describe a deformation retraction of the complement X  C onto the space B described in b). (Hint: describe the deformation in
each 2dim plane that contains the subset D described in b). ) ...
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 Spring '11
 NA
 Topology

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