Term Exam 1

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Term Exam 1 took place on Monday November 7 from 6PM until 8PM at Sidney Smith 1083 (http://www.osm.utoronto.ca/cgi-bin/class_spec/spec03?bldg=SS&room=1083). Here it is:

[edit]

Math 1300Y Topology - Term Exam 1

University of Toronto, November 7, 2005

Solve 4 of the following 5 problems. Each problem is worth 25 points. 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. Let X and Y be topological spaces. A function f:X\to Y satisfies f(\overline{A})\subset\overline{f(A)} for every subset A of X. Prove that f is continuous.

Problem 2. Let \{X_\alpha\}_{\alpha\in A} be a collection of topological spaces. Define "the product topology" on
∏Xα
α
 by its functional properties (not by giving a basis) and prove that if it exists, it is unique.

Problem 3. Let {\mathbb R}^{\mathbb N}_\mbox{box} be the space of sequences of real numbers, taken with the box topology, and let A=\{(a_n)\colon \lim a_n=0\} be the subspace consisting of sequences that converge to 0.

  1. Prove that A is clopen and hence {\mathbb R}^{\mathbb N}_\mbox{box} is not connected.
  2. Find a non-trivial clopen subset B of A and prove that it is indeed clopen. Hence A itself is not connected.

Problem 4. Let X be an arbitrary topological space and let Y be a compact topological space. Prove that the projection \pi_X:X\times Y\to X is a closed map. I.e., show that if F\subset X\times Y is closed, then so is πX(F).

Problem 5. Let an be a sequence of 0s and 1s for which the set A: = {n:an = 1} is infinite. Prove that there is a superlimit Lim for which Lim an = 1. That is, show that there is an element \mu\in\beta{\mathbb N}\backslash{\mathbb N} whose corresponding superlimit Limμ satisfies \mbox{Lim}_\mu\ a_n=1.

Good Luck!

Above the line, edit only if you are sure.
Below the line, edit only lightly.
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Grading Comments

All problems were graded by Dror. Overall, 28 students took the exam; the average grade was 66.82, the median was 70 and the standard deviation was 27.4.

Problem 1.

  1. A general comment on set theory: A = f − 1(B) implies f(A)\subset B, but does not imply f(A) = B.

Problem 2.

  1. A general comment on set theory: given a product
    X = ∏Xα
    α
     there are canonical functions \pi_\alpha:X\to X_\alpha for every α, but no canonical functions i_\alpha:X_\alpha\to X.
  2. It was not required to prove that the product topology exists; only that if it exists, it is unique.

Problem 3.

  1. A^+:=\{(a_n):\lim a_n=0\mbox{ and }\forall n,\,a_n>0\} isn't closed.
  2. If a_n\to 0 and |a_n-b_n|<\frac{1}{n} then b_n\to 0. Facts at this level of complexity need not be proven in a graduate class! Though of course, if you did prove this on your exam, no points were lost...

Problem 4.

  1. In general, \pi_X(F^c)\neq\pi_X(F)^c. Even more in general, "pullback" of sets is a well behaved operation. "Pushforward" is trickier.
  2. The notions "open map" and "closed map" aren't equivalent!

Problem 5.

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Feel free to edit below the line.
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