$$ \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}} \newcommand{\cA}{\mathcal{A}} \newcommand{\cB}{\mathcal{B}} \newcommand{\cC}{\mathcal{C}} \newcommand{\cD}{\mathcal{D}} \newcommand{\cE}{\mathcal{E}} \newcommand{\cF}{\mathcal{F}} \newcommand{\cG}{\mathcal{G}} \newcommand{\cH}{\mathcal{H}} \newcommand{\cI}{\mathcal{I}} \newcommand{\cJ}{\mathcal{J}} \newcommand{\cK}{\mathcal{K}} \newcommand{\cL}{\mathcal{L}} \newcommand{\cM}{\mathcal{M}} \newcommand{\cN}{\mathcal{N}} \newcommand{\cO}{\mathcal{O}} \newcommand{\cP}{\mathcal{P}} \newcommand{\cQ}{\mathcal{Q}} \newcommand{\cR}{\mathcal{R}} \newcommand{\cS}{\mathcal{S}} \newcommand{\cT}{\mathcal{T}} \newcommand{\cU}{\mathcal{U}} \newcommand{\cV}{\mathcal{V}} \newcommand{\cW}{\mathcal{W}} \newcommand{\cX}{\mathcal{X}} \newcommand{\cY}{\mathcal{Y}} \newcommand{\cZ}{\mathcal{Z}} \newcommand{\bA}{\mathbb{A}} \newcommand{\bB}{\mathbb{B}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bD}{\mathbb{D}} \newcommand{\bE}{\mathbb{E}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bG}{\mathbb{G}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bI}{\mathbb{I}} \newcommand{\bJ}{\mathbb{J}} \newcommand{\bK}{\mathbb{K}} \newcommand{\bL}{\mathbb{L}} \newcommand{\bM}{\mathbb{M}} \newcommand{\bN}{\mathbb{N}} \newcommand{\bO}{\mathbb{O}} \newcommand{\bP}{\mathbb{P}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bS}{\mathbb{S}} \newcommand{\bT}{\mathbb{T}} \newcommand{\bU}{\mathbb{U}} \newcommand{\bV}{\mathbb{V}} \newcommand{\bW}{\mathbb{W}} \newcommand{\bX}{\mathbb{X}} \newcommand{\bY}{\mathbb{Y}} \newcommand{\bZ}{\mathbb{Z}} $$
Due: September 20th, 2019

Math 104 Assignment 3

  1. Warm up (not to be submitted)

    Determine if each of the following statements is true or false. Some of these require a bit of thought, and some are almost directly from the lectures. Do not submit this problem.
    1. Every unbounded subset of $\R$ is infinite.
    2. Every infinite subset of $\R$ is unbounded.
    3. If $E$ is a bounded subset of a metric space $X$, then every subset of $E$ is also bounded.
    4. $\Q$ is a dense subset of $\R$.
    5. $\R$ is a dense subset of $\C$, where $\C$ is given the metric $d(a+ib, x+iy) = \sqrt{(a-x)^2+(b-y)^2}$.
    6. If $E$ is a subset of $X$, then $(E^c)^c = E$.
    7. If $E$ is an open subset of a metric space $X$, then $E^c$ is closed.
    8. If $E$ is a subset of a metric space which is not open, then it is closed.
    9. If $(M, d)$ is a metric space, then $M$ is open.
    10. If $(M, d)$ is a metric space, then $M$ is closed.
    11. If $(M, d)$ is a metric space, then $M$ is bounded.
    12. If $E$ is a bounded subset of a metric space, then it is either open or closed.
    13. Any finite subset of a metric space is closed.
    14. Any finite subset of a metric space is open.
    15. Any finite subset of a metric space is compact.
    16. No finite non-empty subset of a metric space is closed.
    17. No finite non-empty subset of a metric space is open.
    18. No finite non-empty subset of a metric space is compact.

  2. De Morgan's Laws

    Verify the following identities, where $(E_\alpha)$ is a collection of subsets of a set $M$.
    1. \[\paren{\bigcap_{\alpha} E_\alpha}^c = \bigcup_\alpha E_\alpha^c\]
    2. \[\paren{\bigcup_{\alpha} E_\alpha}^c = \bigcap_\alpha E_\alpha^c\]

  3. Limit points

    Let $(M, d)$ be a metric space, $E \subseteq M$, and $E'$ be the set of limit points of $E$.
    1. Show that $x \in E'$ if and only if every open set containing $x$ contains infinitely many elements of $E$.
    2. Prove that $E'$ is closed.
    3. Prove that $E$ and $\overline{E}$ have the same limit points.
    4. Prove that $E$ and $E'$ must have the same limit points, or show by example that this need not be the case.

  4. Interiors

    Let $(M, d)$ be a metric space. Recall that the interior of a subset $E \subseteq M$ is defined as \[E^{\intr} = \set{x \in M \mid \exists r \gt 0, B_r(x) \subseteq E}.\]
    1. Show that the interior of any set is open.
    2. Show that $E$ is open if and only if $E = E^\intr$.
    3. Prove that if $G \subseteq E$ is open, then $G \subseteq E^\intr$.
    4. Prove that the complement of the interior of $E$ is the closure of the complement of $E$. That is, show that \[\paren{E^\intr}^c = \overline{E^c}.\]
    5. Prove or provide a counterexample to the following claims:
      1. $\overline{E} = \overline{E^\intr}$.
      2. $E^{\intr} = \paren{\overline{E}}^\intr$.

  5. Compact sets in the discrete metric

    Let $M$ be a set, and $d$ the discrete metric on $M$, which is given by \[d(x, y) = \begin{cases}0 & \text{ if } x = y \\ 1 & \text{ otherwise}\end{cases}.\] Which subsets of $M$ are compact?

  6. Closures of balls versus closed balls

    1. Let $(M, d)$ be a metric space, $x\in M$, and $r \gt 0$. Show that \[\overline{B_r(x)} \subseteq \set{y \in M \mid d(x, y) \leq r}.\]
    2. Give an example to show that this inclusion may be proper. (Hint: consider $\Z$ as a metric space (although there are also many other examples).)

Here are some further problems to think about out of interest. You do not need to attempt them, nor should you submit them with the assignment.