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Useful links

Office hours:

  • Mondays 9:00-10:00
  • Wednesdays 14:00-16:00

Email

GSI:

Rahul Dalal
  • M 10:30-12:30
  • TTh 17:30-19:30
  • WF 11:00-13:00

Exams

Let $(\mathcal M, d)$ be a metric space.

  1. Suppose that $K_1, K_2 \subseteq M$ are compact. Which is the weakest assumption below which guarantees that $K_1 \cup K_2$ is compact?

    1. $K_1\cap K_2 \neq \emptyset$.
    2. $M = \R^n$ for some $n \in \N$.
    3. $K_1 \subseteq K_2$.
    4. No further assumption.

  2. Which of the following is the statement that the sequence $(a_n)_n$ does not converge?

    1. $\forall L \in M \, \forall r \gt 0 \, \exists N \, \forall n \gt N, \, d(a_n, L) \geq r$.
    2. $\exists L \in M \, \exists r \gt 0 \, \forall N \, \exists n \gt N, \, d(a_n, L) \geq r$.
    3. $\forall L \in M \, \exists r \gt 0 \, \forall N \, \exists n \gt N, \, d(a_n, L) \geq r$.
    4. $\exists L \in M \, \forall r \gt 0 \, \exists N \, \forall n \gt N, \, d(a_n, L) \geq r$.

  3. If $E \subseteq M$ is not compact, then no infinite open cover of $E$ has a finite subcover.

    1. True.
    2. False.

  4. If $(x_n)_n$ is a sequence which takes on only finitely many distinct values, it converges.

    1. True.
    2. False.