Math 104 Assignment 7

1. Products and Compactness

   Suppose that $(X, d_X)$ and $(Y, d_Y)$ are non-empty metric spaces.

   1. Suppose that $((x_n, y_n))_n$ is a sequence in $(X\times Y, d_\infty)$. Prove that $((x_n, y_n))_n$ is Cauchy if and only if $(x_n)_n$ and $(y_n)_n$ are both Cauchy.
   2. Prove that $(X \times Y, d_\infty)$ is complete if and only if $X$ and $Y$ are both complete.
   3. Prove that $(X \times Y, d_\infty)$ is totally bounded if and only if $X$ and $Y$ are totally bounded.
   4. Prove that $(X \times Y, d_\infty)$ is compact if and only if $X$ and $Y$ are compact.
   5. Prove that $(X \times Y, d_2)$ is compact if and only if $X$ and $Y$ are compact. (Hint: there may be a homework problem from an earlier assignment that is very useful here.)
   6. Let $X^1 = X$, and for $n \in \N$, let $X^{n+1} = X^n\times X$ with the metric $d_{X^{n+1}}$ defined by \[d_{X^{n+1}} ( (a, x), (b, y) ) = \sqrt{ d_{X^n}(a, b)^2 + d_X(x, y)^2 },\] where $a, b \in X^n$ and $x, y \in X$. Note that $d_{X^{n+1}}$ is the $d_2$ metric corresponding to the product $X^n \times X$. Use induction to show that $X^n$ is compact for every $n \in \N$ if and only if $X$ is compact.

2. An interesting sequence of rational numbers

   Let $(q_n)_n$ be a sequence in $\Q_{\geq0}$ defined as follows: \[q_n = \begin{cases} \frac{a}{b} & \text{ if } n = 2^a3^b \text{ for some }a, b \in \N \\ 0 & \text{otherwise} \end{cases}.\] Show that for any $x \in \R_{\geq0}$ there is a subsequence of $(q_n)$ converging to $x$. (Hint: first show that every positive rational number occurs infinitely often in $(q_n)_n$, and recall that $\Q$ is dense in $\R$.)

3. Subsubsequences

   1. Suppose that $(M, d)$ is a metric space and $a \in M$. Suppose further that $(x_k)_k$ is a sequence in $M$ with the property that every subsequence of $(x_k)_k$ has a further subsequence which converges to $a$. Show that $(x_k)_k$ converges to $a$.
   2. Give an example of a sequence $(y_k)_k$ in some metric space, which does not converge but has the property that every subsequence has a further subsequence which does converge.

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. They are coloured by approximate difficulty: easy/medium/hard.

• Not definitions

  1. We say a sequence $(a_n)_n$ in a metric space $(M, d)$ is glomvergent if for every $\epsilon \gt 0$ there is $L \in M$ and $N \in \N$ so that for every $n \gt N$, $d(a_n, L) \lt \epsilon$.
     1. Are all sequences glomvergent?
     2. Are all convergent sequences glomvergent?
     3. Are all glomvergent sequences convergent?
     4. Does a glomvergent sequence exist?
  2. We say a sequence $(a_n)_n$ in a metric space $(M, d)$ is revergent if there is some $L \in M$ so that for every $N \in \N$ and every $n \gt N$, we have $d(a_n, L) \lt d(a_N, L)$.
     1. Are all sequences revergent?
     2. Are all convergent sequences revergent?
     3. Are all revergent sequences convergent?
     4. Does a revergent sequence exist?
  3. We say a sequence $(a_n)_n$ in a metric space $(M, d)$ is trivergent if there is some $L \in M$ and $N \in \N$ so that for every $n \gt N$, for every $\epsilon \gt 0$, we have $d(a_n, L) \lt \epsilon$.
     1. Are all sequences trivergent?
     2. Are all convergent sequences trivergent?
     3. Are all trivergent sequences convergent?
     4. Does a trivergent sequence exist?
  4. We say a sequence $(a_n)_n$ in a metric space $(M, d)$ is provergent if there is some $L \in M$ so that for every $\epsilon \gt 0$ and $N \in \N$, $n \gt N$ implies $d(a_n, L) \lt \epsilon$.
     1. Are all sequences provergent?
     2. Are all convergent sequences provergent?
     3. Are all provergent sequences convergent?
     4. Does a provergent sequence exist?
  5. We say a subset of a metric space is splopen if it contains all of its interior points.
     1. Are all subsets of metric spaces splopen?
     2. Are all open sets splopen?
     3. Are all splopen sets open?
     4. Does a splopen set exist?
  6. We say a subset of a metric space is fhtagn if it contains none of its limit points.
     1. Are all subsets of metric spaces fhtagn?
     2. Are all open sets fhtagn?
     3. Are all fhtagn sets open?
     4. Does a fhtagn set exist?
  7. We say a subset $U$ of a metric space $(M, d)$ is tropen if for every $x \in M$ and every $r \gt 0$, $B_r(x) \subseteq U$.
     1. Are all subsets of metric spaces tropen?
     2. Are all open sets tropen?
     3. Are all tropen sets open?
     4. Does a tropen set exist?
  8. We say a subset $U$ of a metric space $(M, d)$ is hopen if for every $x \in U$ and every $r \gt 0$, $B_r(x) \subseteq U$.
     1. Are all subsets of metric spaces hopen?
     2. Are all open sets hopen?
     3. Are all hopen sets open?
     4. Does a hopen set exist?
  9. We say a subset of a metric space is posed if it is not open.
     1. Are all subsets of metric spaces posed?
     2. Are all closed sets posed?
     3. Are all posed sets closed?
     4. Does a posed set exist?
  10. We say a subset of a metric space is clopen if it is both closed and open.
      1. Are all subsets of metric spaces clopen?
      2. Are all open sets clopen?
      3. Are all closed sets clopen?
      4. Does a clopen set exist?

• Pseudometrics

  (The hardest part of this problem is understanding the definitions.)

  A pseudometric on a set $M$ is a function $p : M\times M \to \R_{\geq0}$ such that:

  • for every $x \in M$, $p(x, x) = 0$;
  • for every $x, y \in M$, $p(x, y) = p(y, x)$; and
  • for every $x, y, z \in M$, $p(x, z) \leq p(x, y) + p(y, z)$.
  Note that we have not insisted that $p(x, y) = 0$ only if $y = x$.

  An equivalence relation on a set $X$ is a relation which is reflexive, symmetric, and transitive. That is, a relation $\sim$ such that:

  • for every $x \in X$, $x \sim x$;
  • for every $x, y \in X$, if $x \sim y$ then $y \sim x$; and
  • for every $x, y, z \in X$, if $x \sim y$ and $y \sim z$ then $x \sim z$.
  Given $x \in X$, the equivalence class of $x$ is the set $[x]_\sim = \set{y \in X \mid x \sim y}$. Notice that if $x \sim y$ then $[x]_\sim = [y]_\sim$. The set of equivalence classes is denoted \[{X/\!\sim} = \set{[x]_\sim \mid x \in X}.\]

  As an example, fix $k \in \N$ and define a relation $\sim$ on $\Z$ by $n \sim m$ if and only if $k$ divides $n - m$. Then, for example, $k \sim 15k \sim -2k$, and \[{\Z/\!\sim} = \set{[0]_\sim, [1]_\sim, \ldots, [k-1]_\sim}\] contains precisely $k$ distinct equivalence classes. It is possible to do arithmetic on these classes: we can define $[n]_\sim + [m]_\sim = [n+m]_\sim$ and $[n]_\sim[m]_\sim = [nm]_\sim$, and the definition does not depend on the choice of representatives $n \in [n]_\sim$, $m \in [m]_\sim$. This is the beginning of modular arithmetic.

  1. Prove that if $p$ is a pseudometric on $M$, then the relation $\sim$ defined by $x \sim y$ iff $p(x, y) = 0$ is an equivalence relation.
  2. Given $a, b \in M{/\!\sim}$ equivalence classes, let $x \in a$ and $y \in b$ and define $d(a, b) = p(x, y)$. Prove that $d$ does not depend on the choice of $x$ or $y$ here.
  3. Prove that $d$ is a metric on $M{/\!\sim}$.
  4. Let $p : \R^2\times\R^2 \to \R_{\geq 0}$ be defined by $p(x, y) = |x_1 + x_2 - y_1 - y_2|$. Show that $p$ is a pseudometric. Describe $\R^2{/\!\sim}$.
  5. Let $p : \R \times \R \to \R_{\geq 0}$ be defined by $p(x, y) = \inf\set{|x-y-k| \mid k \in \Z}$. Show that $p$ is a pseudometric. Describe $\R{/\!\sim}$.

• Families of pseudometrics

  Recall that in a metric space $(M, d)$, a sequence $(a_n)_n$ converges to a point $a$ if and only if for every open set $U \subseteq M$ with $a \in U$, there is some $N \in \N$ so that for every $n \gt N$, $a_n \in U$.

  Suppose that $M$ is a set, and let $\mathscr{F}$ be a set of pseudometrics on $M$. Let us call a subset $G \subseteq M$ open if for every $x \in M$, there are finitely many $p_1, \ldots, p_k \in \mathscr{F}$ and some $r \gt 0$ so that \[\set{y \in M \mid p_i(x, y) \lt r \text{ for each } i = 1, \ldots, k} \subseteq G.\]

  1. Prove that a set $G \subseteq M$ is open if and only if it is a union of finite intersections of sets of the form \[B_{p_i, r}(x) = \set{y \in M \mid p_i(x, y) \lt r}.\]

  We say a sequence $(a_n)_n$ in $M$ converges to $a \in M$ if for every open set $U \subseteq M$ with $a \in U$ there is some $N \in \N$ so that for every $n \gt N$, $a_n \in U$.

  2. Prove that every sequence in $M$ has at most one limit if and only if for every $x, y \in M$ there is some $p \in \mathscr{F}$ so that $p(x, y) \gt 0$.
  3. Let $\R^\R$ be the set of functions $\R \to \R$, and for each $x \in \R$, set \[p_x(f, g) = \abs{f(x) - g(x)}.\] Let $\mathscr{F} = \set{p_x \mid x \in \R}$. Show that a sequence of functions $(f_n)_n$ in $\R^\R$ converges to some $f \in \R^\R$ if and only if for each $x \in \R$, \[\lim_{n\to\infty} f_n(x) = f(x).\] (In this case, $(f_n)_n$ is said to converge pointwise to $f$.)

• Common subsequences

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

  1. Suppose that $(x_n)_n, (y_n)_n$ are sequences in $M$. Show that there is a common subsequence along which they both converge. (That is, show that there is an increasing sequence $(n_k)_k$ in $\N$ so that $(x_{n_k})_k$ and $(y_{n_k})_k$ both converge.)
  2. Suppose that $f_1, \ldots, f_T$ are sequences in $M$. (We might also write $(f_1(n))_n, (f_2(n))_n, \ldots, (f_T(n))_n$, but the double-indexing gets annoying.) Show that there is a common subsequence along which they all converge. (That is, show that there is an increasing function $n : \N \to \N$ so that $f_1\circ n, \ldots, f_T\circ n$ all converge; if we denote $n(k)$ by $n_k$, we might write $(f_1(n_k))_k, \ldots, (f_T(n_k))_k$ all converge.)
  3. Suppose that $(f_t)_t$ is a sequence of sequences in $M$ (so for each $t \in \N$, $f_t : \N \to M$). Show that there is a common subsequence along which they all converge. (That is, show that there is an increasing function $n : \N \to \N$ so that for all $t \in \N$, $f_t\circ n$ converges.)

• Products and Sequential Compactness

  Suppose $(X, d_X)$ and $(Y, d_Y)$ are non-empty metric spaces. Prove directly, without appealing to Problem 1 above, that $(X\times Y, d_\infty)$ is sequentially compact if and only if $X$ and $Y$ both are.