10-327/Homework Assignment 7

DBN: Classes: 2010-11: 10-327/Navigation Additions to the MAT 327 web site no longer count towards good deed points # 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, path-connectedness 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 Monday-Tuesday is Fall Break, Thursday - Tychonoff and a taste of Stone-Cech, 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 no-where 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, 2PM-5PM, 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 212-213.

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)

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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)

Solution

Here is the solution to HW7. -Kai page1 page2 page3 page4 page5 page6 page7 page8 page9 page10 page11 page12 page13 page14 page15 page16 page17 page18