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$$ \DeclareMathOperator{\span}{span} $$
Due: November 6th, 2020

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.

    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 \[ [x]_\sim = \set{y \in M \mid x \sim y}.\]

    1. 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.)
    2. Suppose that $H \subseteq M$ is connected, and $x \in H$. Show that $H \subseteq [x]_\sim$.
    3. 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).

  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 $\gamma(1) = y$.

    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 $M$. As before, set \[ [x]_{\sim_p} = \set{y \in M | x \sim_p y}.\]

    1. Show that $\sim_p$ is an equivalence relation: that is,
      1. for any $x \in M$, $x \sim_p x$;
      2. for any $x, y \in M$, if $x \sim_p y$ then $y \sim_p x$; and
      3. for any $x, y, z \in M$, if $x\sim_p y$ and $y \sim_p z$ then $x \sim_p z$.

    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 open and connected, then $E$ is path connected.

    It is not true that connected sets are path connected in general. For example, the set \[T = \set{(0, y) \mid -1 \leq y \leq 1} \cup \set{\paren{x, \sin\paren{\frac1x}} \mid x \gt 0} \subseteq \R^2\] can be shown to be connected but not path connected; it's path components are the two separate pieces in the presentation above.

  3. The Squeeze Theorem for functions

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

  4. A derivative

    Let \begin{align*}f : \R &\longrightarrow \R \\ x &\longmapsto \begin{cases}0 & \text{ if } x \in \Q \\ x^2 & \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 $0$! (That is an exclaimation point, not a factorial; $f$ is discontinuous at $1$.))

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.