MATH 4530 Topology. HW 1 Solution
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MATH 4530 Topology. HW 9
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MATH 4530 Topology. Prelim II, Solutions
TAKE HOME
1. Degree
Let S 1 := cfw_e2i C. We define the degree of a continuous map S 1 S 1 as follows. Let x0 S 1 and
let be a path from 1 S 1 to x0 .
(1) Show that if is a generator of 1 (S 1 , 1), then ()
is a g
MATH 4530 Topology. HW 4
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(1)
MATH 4530 Topology. HW 9 solutions
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MATH 4530 Topology. HW 7
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MATH 4530 Topology. HW 6 solutions
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MATH 4530 Topology. HW 10
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MATH 4530 Topology. HW 5
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MATH 4530 Topology. HW 9
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MATH 4530 Topology. HW 4 Solutions
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MATH 4530 Topology. HW 5 solutions
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MATH 4530 Topology. Prelim I Solutions
In class 75min: 2:55-4:10 Thu 9/30.
Problem 1: Consider the following topological spaces:
(1) Z as a subspace of R with the finite complement topology
(2) [0, ] as a subspace of R.
(3) [0, ] cfw_1 as a subspace of R.
MATH 4530 Topology. Practice Problems For Final Part II solutions
(1) (One point compactification) Assume that X is a non-compact connected Hausdorff space in which
every point has a compact neighborhood. Define X 0 := X t cfw_ as a set. You may use the f
MATH 4530 Topology. Practice Problems For Final solutions
Write the proofs in complete sentences.
(1) Apply Theorem 12.6 [L] (See also Remark 12.7 [L], Theorem 72.1 [M]) and compute the fundamental group of RP2 ](Klein Bottle) where ] means the connected
MATH 4530 Topology. HW 2 Solutions
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MATH 4530 Topology. HW 3 Solution
(1) Show that X is a Hausdorff space if and only if the diagonal := cfw_(x, x) | x X X X is closed
with respect to the product topology.
Solution:
: Assume that the diagonal is closed. Consider a, b X with a , b, note tha
MATH 4530 Topology. HW 8 solutions
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MATH 4530 Topology. HW 10 solutions
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MATH 4530 Topology. Prelim I
In class 75min: 2:55-4:10 Thu 9/30.
Problem 1 (45pts): Consider the following topological spaces:
(1) Z as a subspace of R with the finite complement topology
(2) [0, ] as a subspace of R.
(3) [0, ] cfw_1 as a subspace of R.
2
MATH 4530 Topology. HW 6
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MATH 4530 Topology. HW 2
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MATH 4530 Topology. HW 1
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(1)
Fall 2010 MATH 4530 Topology. FINAL EXAM
2hr 30min: 9:00-11:30 Tue 12/14 2010.
Closed Book, 6 problems, total 105 points (5 points bonus!).
(1) (24 pts) True or False. Dont need to explain. Just say True or False. (Correct answer +4, Wrong
answer +0, No a
MATH 4530 Topology. HW 8 solutions
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Topology Math 4530
Solutions to set 2
4. (a) We must verify that (i) For all k Z there exists some S (a, b)
so that x S (ab) and (ii) that given x S (a, b) S (c, d) there is
some S (e, f ) S (a, b) S (c, d) which contains x. It will follow that
cfw_S (a,
Topology Math 4530
Solutions to set 1
4. (a) We will check that O satises the denition for topology.
i) , X O. This is given.
ii) Closure under union. An open ray A = (a, ) is characterized by
the property that x A implies y A whenever x y , AND x A
impli
Math 453 - Prelim 1- Due Thursday Oct. 4, 4:10 pm, 2007 The rules for this exam are the same as for written HW exercises. No outside help, including other texts, people or the web. You are free to consult with the instructor or TA. 1. Let A and B be compa
Math 4530 First Midterm Exam Brief solutions
1. (a) It suces to show that if a and b are in X then there is a path from f (a) to f (b) in f (X ).
Well, as X is path connected, there is a path : [0, 1] X from a to b. As the composition
of two continuous fu