Math 104 Assignment 4

1. Intersections of non-compact sets can be badly behaved

   We saw in lecture that if $\set{K_\alpha}$ is a collection of compact subsets of $M$ so that the interesction of every finite subcollection is non-empty, then the intersection of all the $K_\alpha$ is non-empty.
   1. Show that this may fail if the $K_\alpha$ are merely assumed to be closed.
   2. Show that this may fail if the $K_\alpha$ are merely assumed to be bounded.
   3. Suppose that $\set{K_\alpha \mid \alpha\in\mathcal{I}}$ is a collection of compact subsets of $M$ so that the intersection $K_\alpha\cap K_\beta$ is non-empty for each $\alpha, \beta \in \mathcal{I}$. Show that it may nonetheless be the case that \[\bigcap_\alpha K_\alpha = \emptyset.\]

2. 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 $\mathscr{U}$.

   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.)

3. 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 $U$.

4. 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 be emphasize where we are taking the ball. Explicitly, if $x \in X$, \[B_r^X(x) = \set{y \in X \mid d(x, y) \lt r}, \qquad\qquad\text{while}\qquad\qquad B_r^M(x) = \set{y \in M \mid d(x, y) \lt r}.\] 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 $x \in G$, $B_r^X(x) \subseteq G$.
   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 3.)
   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.)

5. 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 $\sqrt2 \notin \Q$.)

   An earlier version of this problem had the condition "$q \gt 0$" on the set, which of course makes it fail to be closed. This has been removed.

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.

• 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.

• Open double covers

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

• 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 $\iota^{-1}(U) = U \cap M_2$.)

• Intersections of non-compact sets can be badly behaved

  Find an example that shows the statement in problem 1 may fail if the $K_\alpha$ are closed and bounded, but not necessarily compact. (Hint: for this, you will need to work in a metric space other than $\R^n$.)