Term Exam 2
|#||Week of...||Links (edit)|
|1||Sep 12||About, Tue, Thu, Std2Disc|
|2||Sep 19||Tue, Thu, HW1, 14 Sets|
|3||Sep 26||Tue, Thu, Photo|
|4||Oct 3||Tue, Thu|
|5||Oct 10||HW2, Tue, Thu|
|6||Oct 17||Tue, Thu, HW3|
|7||Oct 24||Mon, Tue, Thu|
|8||Oct 31||Tue, Thu, HW4|
|9||Nov 7||TE1, Tue, Thu|
|10||Nov 14||Tue, Thu, HW5|
|11||Nov 21||Tue, Thu|
|12||Nov 28||Tue, Thu, HW6|
|13||Dec 5||Tue, Thu|
|14||Jan 9||Tue, IT83, Thu, HW7|
|15||Jan 16||Tue, Thu|
|16||Jan 23||Tue, HW8, Thu|
|17||Jan 30||Tue, Thu|
|18||Feb 6||TE3, Tue, Thu|
|19||Feb 13||Tue, Thu|
|20||Feb 27||Tue, Thu, HW9|
|21||Mar 6||Tue, Thu, HW10|
|22||Mar 13||Tue, Thu|
|23||Mar 20||Tue, Thu, HW11|
|24||Mar 27||Tue, Thu|
|25||Apr 3||Tue, Thu, HW12|
|26||Apr 10||Tue, Thu|
|Study||Apr 17||Office Hours|
|Exams||Apr 24||Final, PM|
Term Exam 2 took place on Monday December 12 from 10AM until 12PM at Sidney Smith 2127 (http://www.osm.utoronto.ca/cgi-bin/class_spec/spec03?bldg=SS&room=2127). Here it is:
Math 1300Y Topology - Term Exam 2
University of Toronto, December 12, 2005
Solve 4 of the following 5 problems. Each problem is worth 25 points (though they are not of equal difficulty). If you solve more than 4 problems indicate very clearly which ones you want graded; otherwise a random one will be left out at grading and it may be your best one. You have an hour and 50 minutes. No outside material other than stationary is allowed.
Problem 1. Prove: A connected normal space with more than one point is uncountable. It is not a bad idea to use Urysohn's lemma.
Problem 2. Let d be the metric on defined by (we've shown a long time ago that this metric induces the product topology on X).
- Define "complete" and show that (X,d) is a complete metric space.
- Define "totally bounded" and show that the metric space (X,d) is totally bounded.
(Hence is compact, even without using Tychonoff's theorem.)
Problem 3. A collection F of functions on a set X is "pointwise bounded" if for every there is a constant Mx so that for every we have | f(x) | < Mx. Prove that if F is a pointwise bounded collection of continuous functions on a complete metric space X then it is uniformly bounded on some open set. That is, there is some open set U in X and some constant M so that for every and every one has | f(x) | < M.
Hint. Consider and remember Baire.
Problem 4. Prove that if X is pathwise connected then π1(X,b0) is isomorphic to π1(X,b1) whenever b0 and b1 are basepoints in X.
Problem 5. Let satisfy γ(1) = 1, and let be γ regarded as an element of .
- Prove that if γ is even (γ( − z) = γ(z)) then degγ is an even integer.
- Prove that if γ is odd (γ( − z) = − γ(z)) then degγ is an odd integer.
You are free to use lemmas proven in class provided you quote their relevant parts.
All problems were graded by Dror. Overall, 27 students took the exam; the average grade was 82.22, the median was 95 and the standard deviation was 23.83.