# Term Exam 2

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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
E Dec 12 TE2
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14 Jan 9 Tue, IT83, Thu, HW7
15 Jan 16 Tue, Thu
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18 Feb 6 TE3, Tue, Thu
19 Feb 13 Tue, Thu
R Feb 20
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 $X=[0,1]^{\mathbb N}$ defined by $d((x_k),(y_k))=\max\frac1k|x_k-y_k|$ (we've shown a long time ago that this metric induces the product topology on X).

1. Define "complete" and show that (X,d) is a complete metric space.
2. Define "totally bounded" and show that the metric space (X,d) is totally bounded.

(Hence $[0,1]^{\mathbb N}$ 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 $x\in X$ there is a constant Mx so that for every $f\in F$ 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 $f\in F$ and every $x\in U$ one has | f(x) | < M.

Hint. Consider $A_n:=\{x:\forall f\ |f(x)|\leq n\}$ 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 $\gamma:S^1\to S^1$ satisfy γ(1) = 1, and let $\deg\gamma=[\gamma]\in{\mathbb Z}$ be γ regarded as an element of $\pi_1(S^1, 1)={\mathbb Z}$.

1. Prove that if γ is even (γ( − z) = γ(z)) then degγ is an even integer.
2. 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.

Good Luck!

Above the line, edit only if you are sure.

Below the line, edit only lightly.