10-327/Homework Assignment 3

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 19, 20, 21, and 23 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 24 and 26, just to get a feel for the future.

Doing

Solve the following problems from Munkres' book, though submit only the underlined ones: Problems 6, 7 on page 118, and problems 3, 4, 5, 6, 8, 9, 10 on pages 126-128.

Class Photo

Identify yourself in the 10-327/Class Photo page!

Due date

This assignment is due at the end of class on Thursday, October 14, 2010.

Suggestions for Good Deeds

Annotate our Monday videos (starting with Video:  Topology-100927) in a manner similar to (say)  AKT-090910-1, and/or add links to the blackboard shots, in a manner similar to  Alekseev-1006-1. Also, make constructive suggestions to me, Dror and / or the videographer, Qian (Sindy) Li, on how to improve the videos and / or the software used to display them. Note that "constructive" means also, "something that can be implemented relatively easily in the real worlds, given limited resources".

Dror's notes above / Student's notes below

Discussion

• Question about HW3 8(b). I still don't understand why the uniform topology on ${\mathbb {R} }^{\infty }$ is strictly finer than the product topology. If you find any open nbd in uniform topology of any point in ${\mathbb {R} }^{\infty }$ only finitely many component are in the form of $(x-\epsilon ,x+\epsilon )$ because the sequence has infinitely many $0$'s. Can't I just choose these $(x-\epsilon ,x+\epsilon )$ multiply by infinitely many copies of ${\mathbb {R} }$ in the product topology? -Kai
  • Good thought, but there is something wrong in your logic. This though remains your assignment to do, so what I'll write may sound a bit cryptic: Note that in the uniform topology, the $(\pm \epsilon )$ constraint applies also to the $0$'s. Drorbn 18:13, 12 October 2010 (EDT)
    • But once you get the intersection with ${\mathbb {R} }^{\infty }$, those constraints should disappear?