10-327/Homework Assignment 1

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

Read sections 12 through 17 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 18 through 22, just to get a feel for the future.

Solve and submit the following problems. In Munkres' book, problems 4 and 8 on pages 83-84, problems 4 and 8 on page 92, problems 3 and 4 on page 101, and, for extra credit, the following problem:

Problem. Let $X$ and $Y$ be topological spaces and let $A\subset X$ and $B\subset Y$ be subsets thereof. Using only the definitions in terms of continuity of certain functions, show that the topology induced on $A\times B$ as a subset of the product $X\times Y$ is equal to the topology induced on it as a product of subsets of $X$ and of $Y$. You are allowed to use the fact that two topologies ${\mathcal {T}}_{1}$ and ${\mathcal {T}}_{2}$ on some set $W$ are equal if and only if the identity map regarded as a map from $(W,{\mathcal {T}}_{1})$ to $(W,{\mathcal {T}}_{2})$ is a homeomorphism. Words like "open sets" and "basis for a topology" are not allowed in your proof.

Due date. This assignment is due at the end of class on Thursday, September 30, 2010.

Dror's notes above / Student's notes below