Math 104 Assignment 9

1. Uniformly continuous functions have extensions

   Again, suppose that $E \subseteq X$ is dense, and suppose that $f : E \to Y$ is uniformly continuous on $E$. Suppose also that $Y$ is complete.

   1. Show that if $(x_n)_n$ is a Cauchy sequence in $E$, then $(f(x_n))_n$ is Cauchy in $Y$.
   2. Show that if $(x_n)_n$ and $(a_n)_n$ are sequences in $E$ converging to $x \in X$, then $(f(x_n))_n$ and $(f(a_k))_k$ converge in $Y$ to the same limit.
   3. Prove that there is a uniformly continuous function $\tilde f : X \to Y$ so that $\tilde f(x) = f(x)$ for all $x \in E$. (This function is called a continuous extension of $f$; by the previous problem, we know it is unique.)
   4. Show that if $f$ is merely assumed to be continuous, it may not have a continuous extension.

2. Connected sets

   Let $M$ be a metric space.

   1. Let $x \in M$. Prove that $\set{x}$ is connected.
   2. Suppose that $A, B \subseteq M$ are connected and $A \cap B \neq \emptyset$. Prove that $A \cup B$ is connected.

   Let us define a relation $\sim$ on $M$ as follows: given $x, y \in M$, $x \sim y$ if and only if there is a connected set $A \subseteq M$ with $x, y \in A$. Let us also write \[ [x]_\sim = \set{y \in M \mid x \sim y}.\]

   3. Prove that $\sim$ has the following three properties:
      1. for any $x \in M$, $x \sim x$;
      2. for any $x, y \in M$, if $x \sim y$ then $y \sim x$; and
      3. for any $x, y, z \in M$, if $x\sim y$ and $y \sim z$ then $x \sim z$.
      (This means that $\sim$ is an equivalence relation.)
   4. Suppose that $H \subseteq M$ is connected, and $x \in H$. Show that $H \subseteq [x]_\sim$.
   5. Suppose $x \in M$. Show that $[x]_\sim$ is connected.

   The equivalence classes $[x]_\sim \subseteq M$ are called the connected components of $M$; they are in a loose sense the "largest connected pieces" of $M$. Notice that \[M = \bigcup_{x \in M} [x]_\sim,\] that is, $M$ is the union of its connected components. Also, since $[x]_\sim = [y]_\sim$ whenever $x \sim y$, the connected components of $M$ are disjoint. Since every point of $M$ is in some connected component, it follows that a non-empty space $M$ is connected if and only if it has precisely one connected component.

   To give a few examples, the connected components of $\Z$ are the sets $\set{k}$ for each $k \in \Z$, and the connected components of $\Q$ are the sets $\set{q}$ for each $q \in \Q$. The connected components of $[0, 1) \cup \set{4} \cup (6, 9)$ are $[0, 1)$, $\set4$, and $(6,9)$. The circle $\set{(x, y) \in \R^2 \mid x^2 + y^2 = 1}$ has one connected component, itself; the same is true of any connected set. The empty set has no connected components (since in our definition, the empty set is not connected; we defined it this way so that we could unambiguously list the connected components of a set).

3. Distances and compact sets

   Suppose that $(M, d)$ is a metric space.

   1. Let $d_2$ be the metric on $M \times M$ given by \[d_2( (x_1, x_2), (y_1, y_2) )^2 = d(x_1, y_1)^2 + d(x_2, y_2)^2.\] Prove that \[d : M\times M \to \R_{\geq0}\] is continuous with respect to $d_2$.
   2. Suppose that $J, K \subseteq M$ are compact and non-empty. Prove that there are $x \in J$, $y \in K$ so that \[d(x, y) = \inf\set{d(s, t) \mid s \in J, t \in K}.\] (Any two compact sets have a pair of points which are as close as possible.)

4. The Squeeze Theorem for functions

   Suppose that $E \subseteq M$ and $p$ is a limit point of $E$. Suppose further that $f, g, h : E \to \R$ are functions, and that for some $\delta \gt 0$ and all $x \in E$ with $0 \lt d(x, p) \lt \delta$ we have \[f(x) \leq g(x) \leq h(x).\] Finally, suppose that \[\lim_{x\to p} f(x) = L = \lim_{y \to p} h(y).\] Prove that \[\lim_{t\to p} g(t) = L.\]

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.

• More on connected components

  1. Prove that if $E \subseteq M$ is connected, then so is $\overline E$.
  2. Prove that if $C \subseteq M$ is a connected component of $M$, then $C$ is closed.
  3. Prove that if $M$ has a finite number of connected components, then each component is clopen.
  4. Suppose that $M$ has an infinite number of connected components. Which of the following are possible? Provide examples of those that are, and proofs that the rest are impossible.
     1. All the connected components of $M$ are open.
     2. All but finitely many connected components of $M$ are open.
     3. Infinitely many connected components of $M$ are open, and infinitely many are not.
     4. Finitely many connected components of $M$ are open, and the rest are not.
     5. None of the connected components of $M$ are open.

• Equivalent definitions of connectedness

  Let $M$ be a metric space and $E \subseteq M$. Show that the following are equivalent:

  • $E$ is non-empty, and there do not exist $U_1, U_2 \subseteq M$ disjoint open sets so that $U_1 \cap E$, $U_2 \cap E$ are non-empty with $E \subseteq U_1\cup U_2$;
  • every open cover of $E$ by disjoint sets has a subcover containing a single open set.

• Temperatures

  Show that there are two points on opposite sides of the equator with the same temperature as each other.

  What assumptions are needed to make this true? Are they physically reasonable?

• Mountains

  Suppose that you climb a mountain one day, and descend it the next. Show that there must be some time when your elevation was the same on both days.

• More on derivatives

  Suppose that $p \in \R$, and that $E \subseteq \R$ has the property that $p \in E' \cap (E^c)'$. Suppose further that $f, g : \R \to \R$. Define \begin{align*} \varphi : \R &\longrightarrow \R \\ t &\longmapsto \begin{cases} f(t) &\text{ if } t \in E \\ g(t) &\text{ if } t \in E^c.\end{cases} \end{align*} Prove that $\varphi$ is differentiable at $p$ if and only if $f$ and $g$ both are, $f'(p) = g'(p)$, and $f(p) = g(p)$.