Date
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WORK DUE at the beginning of class or lab
unless otherwise noted!
Be sure to follow the
Proof-Writing Samples and the
Proof-Writing Checklist
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| Dec 16 - Tues |
- Final Project Presentations 3-5:30pm
- Topology of the Universe by Fabien Dass
- Euler's Formula and Topological Invariants by Tiffney Duke
- Algebraic Topology by John Foley
- Topology of the Internet by Lindsay Lamb
- Knots by Natahsa Mabe
- Topology and Economics by Ryan Nichols
- Topology and Electric Circuit Design by Jonathan Watson
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| Fri - Dec 12 |
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| Dec 9 - Tues |
- PS 9 due at 5pm
For each of the following, if the set is not compact, then
produce an open cover that has no finite subcover. Otherwise, prove
that it is compact: [0,1), {1/n s.t. n is a natural number},
[0,infinity), X with the cofinite topology.
p. 131 number 11
Prove that the finite
union of two compact sets in a space X is compact in X.
Extra Credit
Show that the intersection of two compact sets in a Hausdorff
space X is compact in X.
Extra Credit
Show that Hausdorff is required in the above statement, ie
that the intersection of compact subspaces of a space X is not
necessarily
compact as follows:
Look at Y=[0,1] U [2,3] with the equivalence relation ~ on Y s.t.
t ~ t for all t,
t ~ t+2 for all t in [0,1),
t ~ t-2 for all t in [2,3)
Show that Y/~ is not Hausdorff
Show that [0,1] U [2,3) is compact in Y/~
Show that [0,1) U[2,3] is compact in Y/~
Show that the intersection of these two compact sets in Y/~
is not compact in Y/~
Extra Credit
Prove that If X is compact Hausdorff under both T and T', then T=T' or
they are not comparable.
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| Dec 2 - Tues |
- PS 8
Using ideas of connected spaces, show that no pair of the following
is homeomorphic: (0,1), (0,1], [0,1]
Using ideas of connected spaces, show that R^2 and R
are not homeomorphic
Show why each of the following is or is not connected:
R_l and R_zar=R_fc
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| Nov 25 - Tues |
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| Nov 20 - Thur |
- Problem Set 7 is due by 5pm (Hints will be posted on WebCT)
2.6 1, 9, 17b.
(Grad) just the closed part of 5
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| Nov 18 - Tues |
- Go over Problem Set 6 Solutions and take WebCT quiz 4 (open notes).
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| Nov 13 - Thur |
- Test up to and including Hausdorff
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| Nov 11 - Tues |
- Problem Set 6 due by 5pm
Prove that X is discrete iff every function f : X-->R is continuous
p. 57 #20
Prove or Disprove that the following are homeomorphic
a) S^1 and {(x,y) | max(|x|,|y|) = 1}, both with the subspace topologies
of R^2.
b) R with the standard topology and R_cf with the finite complement
topology.
c) (Extra Credit) [1,2) and {0}U(1,2) with the subspace topologies
of R.
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| Nov 6 - Thur |
- Skim 1.7, 2.1 and 2.2
- Work on PS 6
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| Nov 4 - Tues |
- Self-Evaluation for oral test due. Reflect on your oral
presentations. What are aspects of your presentations that went
especially well? How about aspects that could use improvement?
Give yourself a grade.
- Work on PS 6
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| Oct 30 - Thur |
- Go over PS 5 Solutions and compare them with your work.
Then take WebCT quiz 3.
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| Oct 28 - Tues |
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Test 2 on examples and definitions up through and including Problem Set 4.
Oral test continued. Be able to answer your questions and
also be able to recite any related definitions.
Both are closed to notes.
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| Oct 21 - Tues |
- Problem Set 5
Which of the following are Hausdorff? (Informally justify your answers.)
a) X={1,2,3} with the topology={Empty set,
{1,2}, {2},{2,3},{1,2,3}}
b) The discrete topology on R
c) The Cantor
Set with the subspace topology induced as a subset of the usual
topology on R
d) Rl, the lower limit topology on R
e) The product topology Rl x R
Prove that X is Hausdorff implies Delta={(x,x) | x in X} is closed in XxX
(Grad)
Delta={(x,x) | x in X} is closed in XxX implies that X is Hausdorff
p. 159 14 b and c
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| Oct 16 - Thur |
- Go over PS 4 Solutions and compare them with your work.
Then take WebCT quiz 2 on material up through and including Project 4.
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| Oct 14 - Tues |
- Oral test (closed to notes) on material up through and including Project 3.
Be sure that you could explain why each answer is true or false from
WebCT quiz 1, and that you can explain the answers to test 1.
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| Oct 13 - Mon |
- Try 2 of WebCT quiz 1 - see bulletin board message before
retaking it.
- By today, be sure that you have
gotten in to see me in office hours for a 10 minute conference.
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| Oct 9 - Thur |
- Problem Set 4 p. 34-35
4, 5, 14, 18
(Grad) 12 and 17
Note: On 17 and 18 give
informal justifications instead of formal proofs.
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| Oct 2 - Thur |
- Test on material up through and including Problem Set 3.
Focus on definitions, examples, history, and big picture understanding
from class notes and problem set solutions.
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| Sept 30 - Tues |
- Take the WebCT quiz. Notes are allowed on this.
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| Sep 25 - Thur |
- Problem Set 3 p. 24-26 DUE at 5pm
5 the first part, 7, 17, 20, 21
(Grad) 5 the second part about whether Tau is the lower limit
topology on R.
(Grad) 18
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| Sep 18 - Thur |
- Read p. 15-19,
read through PS 2 solutions and begin working on Problem Set 3.
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| Sept 16 - Tues |
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| Sept 11 - Thur |
- Memorize the definitions of Bd(x,E) and U open in (X,d).
- Re-read the proof on page 8.
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| Sept 9 - Tues |
- Read Patty Section 1.1 and 1.2. Begin working on Problem Set 2.
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| Sept 2 - Tues |
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| Aug 28 - Thur |
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- Read handouts given during Tuesday's class. Begin working on
Problem Set 1.
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