$$ \newcommand{\cis}{\operatorname{cis}} \newcommand{\norm}[1]{\left\|#1\right\|} \newcommand{\paren}[1]{\left(#1\right)} \newcommand{\sq}[1]{\left[#1\right]} \newcommand{\abs}[1]{\left\lvert#1\right\rvert} \newcommand{\set}[1]{\left\{#1\right\}} \newcommand{\ang}[1]{\left\langle#1\right\rangle} \newcommand{\floor}[1]{\left\lfloor#1\right\rfloor} \newcommand{\ceil}[1]{\left\lceil#1\right\rceil} \newcommand{\C}{\mathbb{C}} \newcommand{\D}{\mathbb{D}} \newcommand{\R}{\mathbb{R}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\N}{\mathbb{N}} \newcommand{\F}{\mathbb{F}} \newcommand{\T}{\mathbb{T}} \renewcommand{\S}{\mathbb{S}} \newcommand{\intr}{{\large\circ}} \newcommand{\limni}[1][n]{\lim_{#1\to\infty}} \renewcommand{\Re}{\operatorname{Re}} \renewcommand{\Im}{\operatorname{Im}} $$

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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 $E \subseteq M$, and $x \in E$ is not a cluster point of $E$. Then the statement "$x$ is an interior point of $E$"...
    1. ...is always true.
    2. ...can be either true or false.
    3. ...is never true.
  2. Let $E \subseteq M$, and suppose $E$ is not closed. Then $\overline{E}$ is not open.
    1. True.
    2. There is at least one example where this holds, and at least one where it fails.
    3. There are no examples where this is true.
  3. Suppose $E \subseteq M$. Then $E^\circ \subseteq \overline{E}$. (I.e., the interior of $E$ is a (not necessarily proper) subset of the closure of $E$.)
    1. True.
    2. There is at least one example where this holds, and at least one where it fails.
    3. There are no examples where this is true.
  4. Let $S = (-\infty, 0) \cup [1, 2) \cup \set{3 + \frac1n\mid n \in \N} \subset \R$. Which of the following is not a limit point of $S$?
    1. $0$
    2. $1$
    3. $\frac32$
    4. $3+\frac1{10000}$