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 100, 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.

Note on Question 8, Page 92. (Added 8:00AM, September 27). One should think that "describe" for verbal things is like "simplify" for formula-things. The topologies in question were given by a verbal description; the content of the question is that you should be giving a simpler one, and the best is if it is of the form "the topology in question is the trivial topology", or something like that. Note that the resulting topology may also depend on the direction of the line $L$, so you may wish to divide your answer into parts depending on that direction.

Dror's notes above / Student's notes below

• Dror do you mean question 3 and 4 on page 100? There is no question 3 and 4 on page 101. -Kai
  • Sorry and thanks for the correction, indeed I meant page 100. (BTW, next time you sign a wiki submission, use "~~~~" (four "tilde" symbols) and see what it does). Drorbn 18:03, 26 September 2010 (EDT)

• For question 8, page 92, are we only to consider horizontal and vertical lines, or lines in any direction? (I realize this question is slightly foolish, but I don't understand why else the question would specify that we consider a "straight line" -- aren't all lines straight?) Anai 19:47, 28 September 2010 (EDT)Ana