10-327/Classnotes for Thursday November 18

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_327-101118-142430.jpg (unfortunately some blackboards from the middle of the proof or Urysohn's lemma got forgotten).

Dror's notes above / Student's notes below

Here are some lecture notes:

Lecture part 1

Lecture part 2

Lecture part 3

Lecture part 4

Questions by Kai Xwbdsb 21:26, 19 November 2010 (EST):

1. I have concern about this "Adding T_1 thing". When we are proving the Urysohn's Lemma, we proved that if X is a T.S. and A,B are any two disjoint closed sets in X, they can be separated by two disjoint open sets in X iff they can be separated by a continuous function. So there is no T_1 axiom involved in any of these two proofs. So Urysohn's Lemma is something more general than T_4 iff T_4.5 right?
   • You are probably right and Urysohn applies even without $T_{1}$. Yet we want $T1$ all else $T_{4.5}$ does not imply $T_{3.5}$ and below, and we will need $T_{3.5}$ very soon. Drorbn 06:02, 20 November 2010 (EST)
2. I just want to also confirm the definition of T_4 and T_4.5: T_4 is T_1 + separation by two disjoint open sets for any two disjoint closed sets and T_4.5 is T_1 + separation by a continuous function for any two disjoint closed sets?
   • Indeed so. Drorbn 06:02, 20 November 2010 (EST)
3. The only purpose of this manually adding T_1 thing is that so that T_4 and T_4.5 could imply T_3 T_2 T_1 right?
   • Indeed. Drorbn 06:02, 20 November 2010 (EST)
4. Dror you said before that induction can go up to finite n but why in this case induction will work for a countably infinite set of elements?
   • We are never making a statement about an infinite set of rational numbers directly through induction. We are only constructing something for the $n$'th rational, for each $n$. We then look at the totality of these constructions, but that's already outside the induction. Drorbn 06:02, 20 November 2010 (EST)

• • So you mean if properly used, mathematical induction could prove things involve countably infinite objects right? Just like in this case. -KaiXwbdsb 11:32, 20 November 2010 (EST)