Math 104 Assignment 4

1. 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. $E^{\intr} = \paren{\overline{E}}^\intr$.
   6. Let $E = \big(\Q\cap (0, 2)\big) \cup [1, 3] \cup \set5 \subset \R$. Determine $E^\circ$.

2. Relatively open sets

   Suppose that $(M, d)$ is a metric space, and $X \subseteq M$. Recall that $X$ is therefore a metric space with metric inherited from $M$: for $x, y \in X$, we have $d_X(x, y) = d(x, y)$. However, the open balls in $X$ are different from those in $M$. We emphasize this by writing $B_r^X(x)$ or $B_r^M(x)$ to indicate where we are taking the ball. Explicitly, if $x \in X$, \[B_r^X(x) = \set{y \in X \mid d_X(x, y) \lt r}, \qquad\qquad\text{while}\qquad\qquad B_r^M(x) = \set{y \in M \mid d(x, y) \lt r}.\] It also follows that we have different notions of "open" between $X$ and $M$: if $G \subseteq X$, we say $G$ is open relative to $X$ if for every $x \in G$ there is $r \gt 0$ so that $B_r^X(x) \subseteq G$.
   1. Show that if $U \subseteq M$ is open relative to $M$, then $U \cap X$ is open relative to $X$. (Hint: notice that for $x \in X$, $B_r^X(x) = B_r^M(x) \cap X$.)
   2. Show that if $G \subseteq X$ is open relative to $X$, then there is some $U \subseteq M$ open relative to $M$ so that $G = U \cap X$. (Hint: use problem 1.)
   3. Give an explicit example of a metric space $M$ and sets $G \subseteq X \subseteq M$ so that $G$ is open relative to $X$ but not relative to $M$. (Hint: take $X$ to be a subset of $M$ which is not open.)
   4. Suppose $K \subseteq X$. Prove that $K$ is compact relative to $X$ if and only if it is compact relative to $M$. (Hint: use parts A and B to convert open covers relative to $M$ to open covers relative to $X$ and vice versa.)

3. Limit Cluster points

   Let $(M, d)$ be a metric space, $E \subseteq M$, and $E'$ be the set of limit cluster 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 $x \in \overline{E}$ if and only if for every $r \gt 0$, $B_r(x) \cap E \neq \emptyset$.

4. 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?

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.

• Warm up

  Determine if each of the following statements is true or false.

  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. If $E$ is an unbounded subset of a metric space $X$, then every subset of $E$ is also unbounded.
  5. $\Q$ is a dense subset of $\R$.
  6. $\R$ is a dense subset of $\C$, where $\C$ is given the metric $d(a+ib, x+iy) = \sqrt{(a-x)^2+(b-y)^2}$.
  7. If $E$ is a subset of $X$, then $(E^c)^c = E$.
  8. If $E$ is an open subset of a metric space $X$, then $E^c$ is closed.
  9. If $E$ is a subset of a metric space which is not open, then it is closed.
  10. If $(M, d)$ is a metric space, then $M$ is open.
  11. If $(M, d)$ is a metric space, then $M$ is closed.
  12. If $(M, d)$ is a metric space, then $M$ is bounded.
  13. If $(M, d)$ is a metric space, then $M$ is unbounded.
  14. If $E$ is a bounded subset of a metric space, then it is either open or closed.
  15. Any finite subset of a metric space is closed.
  16. Any finite subset of a metric space is open.
  17. Any finite subset of a metric space is compact.
  18. No finite non-empty subset of a metric space is closed.
  19. No finite non-empty subset of a metric space is open.
  20. No finite non-empty subset of a metric space is compact.

• Open sets are unions of balls

  Show that if $U \subseteq M$ is an open set, then there is a set of balls whose union is $U$.

• Clustrer points, II

  Let $(M, d)$ be a metric space, $E \subseteq M$, and $E'$ be the set of cluster points of $E$.

  1. Show by example that $E$ and $E'$ need not have the same cluster points; that is, show that $E'$ and $(E')'$ may not be equal.
  2. Prove or provide a counterexample to the claim that $(E')' \subseteq E'$.
  3. Prove that $E'$ is closed.
  4. Prove or disprove the following statement: $x \in E'$ if and only if for every $r\gt0$ there is $y \in E$ with $\frac{r}2 \lt d(x, y) \lt r$.
  5. Determine the cluster points of the set $S = \set{3-\frac3{m}\mid m \in \N} \cup [0,2] \subset \R$.

• Distances and compact sets

  (This problem can be solved with what we know now, but will be easier with more tools at our command. It may show up on a later assignment.)
  1. Let $K$ be a non-empty compact subset of a metric space $(M, d)$. For $x \in M$, define \[d(x, K) = \inf\set{d(x, y) \mid y \in K}.\] Show that there is a point $y\in K$ so that $d(x, K) = d(x, y)$. Is this point unique?
  2. Let $(M, d)$ be a metric space. Let $\mathscr{K} := \set{K \subseteq M \mid K \text{ is compact and non-empty}}$. Define \begin{align*} d_H : \mathscr{K}\times\mathscr{K} &\to \R_{\geq 0} \\ (K, F) &\mapsto \max\paren{\sup\set{d(x, K) \mid x \in F}, \sup\set{d(y, F) \mid y \in K}}. \end{align*} Show that $(\mathscr{K}, d_H)$ is a metric space.

• Nepo functions

  Let us say that a function $f : M_1 \to M_2$ between metric spaces $(M_1, d_1)$ and $(M_2, d_2)$ is nepo if the preimage of every open subset of $M_2$ is open in $M_1$. That is, if all sets of the form \[f^{-1}(U) = \set{x \in M_1 \mid f(x) \in U}\] where $U \subseteq M_2$ is open are themselves open, as subsets of $M_1$.
  1. Suppose that $K \subseteq M_1$ is compact and $f : M_1 \to M_2$ is nepo. Show that $f(K) = \set{f(x) \mid x \in K} \subseteq M_2$ is compact.

  2. Suppose that $M_1 \subseteq M_2$ and moreover that $d_1(x, y) = d_2(x, y)$ for all $x, y \in M_1$; that is, $M_1$ is a sub-metric space of $M_2$. Show that the inclusion function $\iota : M_1 \to M_2$ given by $\iota(x) = x$ is nepo.

     (Note that the metric space in which a set is considered is important here! The goal is to show that open subsets of $M_2$ have preimages in $M_1$ which are open as subsets of $M_1$. You may find it useful to first show that $\iota^{-1}(U) = U \cap M_2$.)

• Compactness in the rational numbers

  Let $S = \set{q \in \Q \mid q^2 \lt 2}$. Show that $S$ is closed and bounded as a subset of the metric space $\Q$, but is not compact. Is $S$ open in $\Q$? (Hint: use problem 4, and the fact that $\sqrt2 \notin \Q$.)

• 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).)

• Open double covers

  Let $(M, d)$ be a metric space. Call a collection $\mathscr{U}$ of open sets an open double cover of $E \subseteq M$ if every point $x \in E$ is contained in at least two elements of $\mathscr{U}$.
  1. Show that if $K \subseteq M$ is compact, then every open double cover of $K$ has a finite open double subcover. (That is, there is a finite subset of the original open double cover which remains an open double cover.)
  2. Is it true that if $K \subseteq M$ is such that every open double cover has a finite open double subcover, then $K$ is compact?