10-327/Classnotes for Monday December 6

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 [math]\displaystyle{ {\mathbb R} }[/math], HW4, HW4 Solutions, Thursday - connectedness, path-connectedness and products 7 Oct 25 Monday - Compactness of [math]\displaystyle{ [0,1] }[/math], 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 - [math]\displaystyle{ T_{3.5} }[/math] and [math]\displaystyle{ I^A }[/math], 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_327-101206-142909.jpg.

Video:  Topology-101206

Dror's notes above / Student's notes below

Lecture Notes

• Question. The fact that the metric space of real-valued functions on the unit interval with uniform metric is complete uses the fact that [0,1] is compact right? If the function space is defined on a non-compact topological space is that necessarily complete?... -Kai Xwbdsb 00:01, 20 December 2010 (EST)
  • No, the compactness of [math]\displaystyle{ [0,1] }[/math] is not used. As we said in class, if [math]\displaystyle{ (f_n) }[/math] is Cauchy in the uniform metric, then for any [math]\displaystyle{ x }[/math], the sequence [math]\displaystyle{ (f_n(x)) }[/math] is Cauchy in [math]\displaystyle{ {\mathbb R} }[/math], so it has a limit. Call that limit [math]\displaystyle{ f(x) }[/math]; it is not hard to show that [math]\displaystyle{ f }[/math] is continuous and that [math]\displaystyle{ f_n\to f }[/math]. Theorem 43.6 in Munkres is a slight generalization of this. Drorbn 07:12, 20 December 2010 (EST)

Thanks Dror.

Everybody good luck on the exam!-Kai

Great course! Thank you very much for all your help Dror and all the classmates in this class. -Kai