Due: November 15th, 2019

Math 104 Assignment 9

  1. 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.
    3. Define a relation $\sim$ on $M$ by $x \sim y$ if and only if there is a connected set $A \subseteq M$ with $x, y \in A$. Prove 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 $\sim$ is an equivalence relation, the connected components of $M$ are disjoint. Since every point of $M$ is in some connected component, it is easy to check 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).

  2. Path connected sets

    Suppose $M$ is a metric space. If $x, y \in M$, a path (in $M$) from $x$ to $y$ is a continuous function $\gamma : [0, 1] \to M$ with $\gamma(0) = x$ and $\gamma(1) = y$.

    1. Define a relation $\sim_p$ on $M$ by $x \sim_p y$ if and only if there is a path from $x$ to $y$ in $M$. Show that $\sim_p$ is an equivalence relation.

    The equivalence classes $[x]_{\sim_p} \subseteq M$ are called path components of $M$. If $M$ has exactly one path component, it is called path connected. (Note that $\emptyset$ is not path connected: it has zero path components, not one.)

    1. Show that if $M$ is path connected, then it is connected.

    2. Suppose that $E \subseteq \R^n$ is open. Show that the path components of $E$ are open (in $\R^n$).

      (If you find it useful, you may use without proof the fact that functions of the form \begin{align*}f : \R &\longrightarrow \R^n \\ t &\longmapsto \vec{a} + t\vec{b}\end{align*} are continuous, where $\vec{a}, \vec{b} \in \R^n$.)

    3. Prove that if $E \subseteq \R^n$ is non-empty, open, and connected, then $E$ is path connected.

    It is not true that connected sets are path connected in general. Consider, for example, the set \[T = \set{(0, y) \mid -1 \leq y \leq 1} \cup \set{\paren{x, \sin\paren{\frac1x}} \mid x \gt 0} \subseteq \R^2.\] It can be shown that while $T$ is connected, the two pieces above are distinct path components of $T$.

  3. The Squeeze Theorem for functions

    Suppose that $E \subseteq M$ and $p$ is a limit point of $E$. Suppose further that $f, g, h : E \to Y$ 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 \[\lim_{t\to p} g(t) = L.\]

  4. A derivative

    Let \begin{align*}f : \R &\longrightarrow \R \\ x &\longmapsto \begin{cases}x^2 & \text{ if } x \in \Q \\ 0 & \text{ otherwise}.\end{cases}\end{align*} Show that $f$ is differentiable at $0$, and find its derivative there. (Notice that $f$ is discontinuous everywhere except at $0$!)

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.