10-327/Classnotes for Thursday November 25

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

See some blackboard shots at BBS/10_327-101125-142103.jpg.

Dror's notes above / Student's notes below

Here is a lecture note for today:

Lecture Nov 25

Question

Question. The first half of Tietze's theorem isn't very surprising as a limiting process of approximations. But the second half is just like a magic? I don't understand what has been implicitly used here. The "boundedness" property only depends on the metric we define on a set and it does not have anything to do with topology. We are linking R with (-1,1) with a homeomorphism which is completely not metric-related. And suddenly all the unbounded cts functions all become bounded cts functions?......What has been used here? Did we implicitly redefined the metric? Why it works out so smoothly just like a magic trick?...

-Kai

Kai - your question is too open-ended to have an answer that fits in a few minutes of typing, so I'd rather answer it in person, if you come to my office hours. Drorbn 16:39, 6 December 2010 (EST)

-Picture

One picture summary of what you should know about regular/completely regular/normal/completely normal spaces. -KaiXwbdsb 07:59, 19 December 2010 (EST) http://katlas.math.toronto.edu/drorbn/index.php?title=Image:10-327_review.JPG