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Additions to the MAT 327 web site no longer count towards good deed points

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Week of...

Notes and Links

1

Sep 13

About This Class, Monday  Continuity and open sets, Thursday  topologies, continuity, bases.

2

Sep 20

Monday  More on bases, Thursdsay  Products, Subspaces, Closed sets, HW1, HW1 Solutions

3

Sep 27

Monday  the Cantor set, closures, Thursday, Class Photo, HW2, HW2 Solutions

4

Oct 4

Monday  the axiom of choice and infinite product spaces, Thursday  the box and the product topologies, metric spaces, HW3, HW3 Solutions

5

Oct 11

Monday is Thanksgiving. Thursday  metric spaces, sequencial closures, various products. Final exam's date announced on Friday.

6

Oct 18

Monday  connectedness in ${\mathbb {R} }$, HW4, HW4 Solutions, Thursday  connectedness, pathconnectedness and products

7

Oct 25

Monday  Compactness of $[0,1]$, Term Test on Thursday, TT Solutions

8

Nov 1

Monday  compact is closed and bounded, maximal values, HW5, HW5 Solutions, Wednesday was the last date to drop this course, Thursday  compactness of products and in metric spaces, the FIP

9

Nov 8

MondayTuesday is Fall Break, Thursday  Tychonoff and a taste of StoneCech, HW6, HW6 Solutions

10

Nov 15

Monday  generalized limits, Thursday  Normal spaces and Urysohn's lemma, HW7, HW7 Solutions

11

Nov 22

Monday  $T_{3.5}$ and $I^{A}$, Thursday  Tietze's theorem

12

Nov 29

Monday  compactness in metric spaces, HW8, HW8 Solutions, Thursday  completeness and compactness

13

Dec 6

Monday  Baire spaces and nowhere differentiable functions, Wednesday  Hilbert's 13th problem; also see December 2010 Schedule

R

Dec 13

See December 2010 Schedule

F

Dec 20

Final exam, Monday December 20, 2PM5PM, at BR200

Register of Good Deeds

Add your name / see who's in!

See Hilbert's 13th


Reading
Read sections $\{31,32,33\}$ in Munkres' textbook (Topology, 2nd edition). Remember that reading math isn't like reading a novel! If you read a novel and miss a few details most likely you'll still understand the novel. But if you miss a few details in a math text, often you'll miss everything that follows. So reading math takes reading and rereading and rerereading and a lot of thought about what you've read. Also, preread sections $\emptyset$, just to get a feel for the future.
Doing
Solve and submit the following problems from Munkres' book:
 Problem 1 on page 199.
 Problem 1 on page 205.
 Problems 1, 4, 5, 8, 9 on pages 212213.
Remark. The following fact, which we will prove later, may be used without a proof: If $X$ is a topological space and $f_{n}:X\to [0,1]$ are continuous functions, then the sum $f(x):=\sum _{n=1}^{\infty }{\frac {f_{n}(x)}{2^{n}}}$ is convergent and defines a continuous function on $X$.
Due date
This assignment is due at the end of class on Thursday, November 25, 2010.

Dror's notes above / Student's notes below


 Question: In problem 1 p205, is asks us to show that any closed subspace of a normal space is also normal. Do we really need the condition that the subspace be closed?  Jdw
 Yes. Drorbn 19:14, 19 November 2010 (EST)
Questions by Kai Xwbdsb 21:26, 19 November 2010 (EST) were moved to Classnotes for Thursday November 18 as they are about that class and not about this assignment. Drorbn 06:03, 20 November 2010 (EST)
 Question. If we have a finite set of continuous function mapping from any topological space into the reals. Any linear combination of these continuous function is still continuous right? The proof is a little extension of 157 proof. This is used to prove the statement you mentioned above. KaiXwbdsb 17:14, 20 November 2010 (EST)
 Any linear combination of functions from ${\mathcal {C}}(X,\mathbb {R} )$ (an uncountable set unless X is empty) is continuous. On its own, however, this proves nothing about infinite sums. Bcd 22:32, 21 November 2010 (EST)
 Question about 9. Is J any indexing set? Possibly uncountable? in the hint: A means any closed set? Kai Xwbdsb 22:13, 20 November 2010 (EST)
 Yes, $J$ is arbitrary and $A$ is closed. Drorbn 06:41, 22 November 2010 (EST)