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MATH 527 Homework 1
September 7, 2007
Problem 1. Let X be a topological space, and A be its subset. Is it true that A , the set of limit points of A is always closed in X? It is not true in general. For a counterexample, we can take X
1 1.1
Point-Set Topology Sufficient Conditions
The following are sufficient conditions to guarantee each property. Normal. X is metrizable. X is compact Hausdorff. X is regular and 2nd countable. X is regular and Lindelf. o Regular. X is a sub
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MATH 527 Homework 11
November 28, 2007
Problem 1. Calculate the homology groups of S 3 with two unlinked circles deleted.
Call our space X and the two deleted circles A and B. We can view a space homoeomorphic to the one above, by se
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MATH 527 Homework 12
December 7, 2007
Problem 1. Show that S n is not homeomorphic to any proper subspace of itself.
Recall invariance of domain: Let U be open in n (or we could use S n ). Let f : U n be continuous and injective. Th
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MATH 527 Homework 10
November 28, 2007
Problem 1. 2 (a) Compute the homology groups of the n-fold torus Tn . (b) Compute the homology groups of the m-fold projective plane
2 RPm.
2 (a) There is only one connected component, so H0 (T
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MATH 27 Homework 9
November 9, 2007
Problem 1. Let p : E B be a covering map. Assume E is path connected and B is simply connected. Show that p is a homeomorphism.
Since E is path connected, the lifting correspondence : 1 (B, b0 )
Serge Ballif
MATH 527 Homework 8
November 2, 2007
Problem 1. (a) Show that every continuous map f : P 2 S 1 is nulhomotopic. (b) Find a continuous map of the torus into S 1 that is not nulhomotopic.
R
(a) Suppose that f is homotopic to a nontri
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Problem 1. Let K = {(x, y, z) group of X = 3 - K.
MATH 527 Homework 7
October 26, 2007
R
R3 | x2 + y2 = 1, z = 0}, the standard circle in R3. Compute the fundamental
X deformation rectracts onto a space X1 which is sphere centered
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MATH 527 Homework 5
October 5, 2007
Problem 1. Show that if A is a nonsingular 3 by 3 matrix having nonnegative entries, then A has a positive eigenvalue.
Consider the closed positive octant P of the sphere S 2 . Since A has nonnegat
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MATH 527 Homework 4
September 28, 2007
Problem 1. Show that every locally compact Hausdorff space is completely regular.
Let X be locally compact Hausdorff. Then X satisfies the conditions necessary for a one-point compactification.
Serge Ballif
MATH 527 Homework 3
September 21, 2007
Problem 1. Let X be a compact Hausdorff space. Let K1 K2 . . . be a sequence of closed connected subsets of X. Let K = Ki . Show that K is connected. i=1 We note first that X is normal since i
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MATH 527 Homework 2
September 14, 2007
Problem 1. Consider a map h : R R , h(x1 , x2 , x3 , . . . ) = (x1 , 4x2 , 9x3 , . . . ). (a) Is h continuous, when R is given by the product topology? (b) Is h continuous, when R is given by th
Serge Ballif
MATH 527 Homework 6
October 12, 2007
Problem 1. Does the Borsuk-Ulam theorem hold for the torus? In other words, for every map f : S 1 S 1 must there exist (x, y) S 1 S 1 , such that f (x, y) = f (-x, -y)?
R2
No. We can imbed th