Date

WORK DUE at the beginning of class or lab
unless otherwise noted!
Be sure to follow the
ProofWriting Samples and the
ProofWriting Checklist

 
Dec 16  Tues 
 Final Project Presentations 35:30pm
 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

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 ~ t2 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.

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

Nov 25  Tues 

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

Nov 18  Tues 
 Go over Problem Set 6 Solutions and take WebCT quiz 4 (open notes).

Nov 13  Thur 
 Test up to and including Hausdorff

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.

Nov 6  Thur 
 Skim 1.7, 2.1 and 2.2
 Work on PS 6

Nov 4  Tues 
 SelfEvaluation 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

Oct 30  Thur 
 Go over PS 5 Solutions and compare them with your work.
Then take WebCT quiz 3.

Oct 28  Tues 

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.

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

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.

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.

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.

Oct 9  Thur 
 Problem Set 4 p. 3435
4, 5, 14, 18
(Grad) 12 and 17
Note: On 17 and 18 give
informal justifications instead of formal proofs.

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.

Sept 30  Tues 
 Take the WebCT quiz. Notes are allowed on this.

Sep 25  Thur 
 Problem Set 3 p. 2426 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

Sep 18  Thur 
 Read p. 1519,
read through PS 2 solutions and begin working on Problem Set 3.

Sept 16  Tues 

Sept 11  Thur 
 Memorize the definitions of Bd(x,E) and U open in (X,d).
 Reread the proof on page 8.

Sept 9  Tues 
 Read Patty Section 1.1 and 1.2. Begin working on Problem Set 2.

Sept 2  Tues 

Aug 28  Thur 

 Read handouts given during Tuesday's class. Begin working on
Problem Set 1.
