10-327/Homework Assignment 8

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

Change in Plans

Time to rest, a bit. For the lack of fun questions, I've decided to postpone this assignment a bit, and to cancel assignment number 9. So this assignment will only be written on Monday November 29, and will be due on the following Monday, December 6.

Reading

Read sections 35, 43, and 45 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 section 48 and 49, just to get a feel for the future. For fun alone, read also section 44.

Doing

Solve and submit the following problems from Munkres' book:

• Problems 2, 5a on pages 223-224.
• Problems 1, 4, 5 on page 270 (the relevant material will be covered on Thursday December 2).
• Problem 1 on page 280 (same comment).

In addition, solve and submit the following problem:

Problem IX. Show that every metric space $X$ can be embedded in the cube $I^{X}$.

Hint. Given a point $x\in X$, what real-valued function on $X$ immediately comes to mind?

Due date

This assignment is due at the end of class on Wednesday, December 8, 2010.

Dror's notes above / Student's notes below

Discussion

• Then how do we count the homework in the final grade? I remember the old rule is to select the best 7 assignments from the total of 9. - Jolin
  • The rule was "worst two assignments will not count", see 10-327/About This Class. Drorbn 05:40, 27 November 2010 (EST)

Question. For question 4. It is to show the intersection is nonempty by creating a Cauchy sequence from these A_n's. But I can't think of a way to prove the other direction. Are we supposed to create a nested sequence of closed sets for every Cauchy sequence? Then how do we ensure closeness? -Kai

Take the closure of each set, as your set. -Jdw Yea thanks Jason. It works because the diam of the closure of a set is the same as the set in the metric topology. -Kai