Geometry of the Line and the Plane

Prove that the mapping $F$ given by (3.1) is a homeomorphism.

Formula (3.1): $F: \mathbb {R}^{*} \rightarrow [-1,1]$, where $\mathbb {R}^{*} = \mathbb {R} \cup \left\{ -\infty , +\infty \right\}$ is the extended real line, is defined by

Let $\mathcal{T}$ consist of all sets of the form $(a, b),[-\infty , b),(a,+\infty ]$ and any union of such sets. Prove that $\mathcal{T}$ is a topology on $[-\infty ,+\infty ]$.

Suppose that $A \subset \mathbb {R}$ is a union of nondegenerate closed intervals. Prove that $A$ can be represented as a countable disjoint union of intervals.

Prove that $\overline{\mathbb {R}}$ is a compact set in the topology described in Exercise 3.1.2.

Prove that $\overline{\mathbb {R}}$ is complete in the topology described in Exercise 3.1.2.

Let $f:[0, \infty ) \rightarrow \mathbb {R}$ be a continuous function such that $\lim_{x \rightarrow \infty } f(x)= L$, and $L<f(0)$. Prove that $f$ attains a maximum value in $[0, \infty )$.

Let $\mathcal{F}$ be the collection of finite unions of intervals of $\mathbb {R}$. Prove that $\mathcal{F}$ is closed under complementation.

Let $\mathcal{F}$ be the set of all countable subsets of $\mathbb {R}$ and suppose that $E \subset \mathbb {R}$ but $E \notin \mathcal{F}$. Let $A:=\left\{ x \in \mathbb {R}:(-\infty , x) \cap E \in \mathcal{F}\right\}$ and $B:= \left\{ x \in \mathbb {R}:(x, \infty ) \cap E \in \mathcal{F}\right\}$. Prove that $A \cap B=\varnothing$ and $A \cup B \neq \mathbb {R}$.

Let $\left\{ A_{i}: i \in I\right\}$ be a collection of closed intervals in $\mathbb {R}$. Prove that the set

is countable.

Let $\mathcal{F}$ be the collection of all closed subsets of $\mathbb {R}^{n}$. Prove that $\mathcal{F}$ has the same cardinality as $\mathbb {R}$.

Prove that every open set in $\mathbb {R}^{n}$ is a countable union of open balls.

Prove that every element of the Cantor set can be written in a unique way as $\sum_{n=1}^{\infty } a_{n} / 3^{n}$, where each $a_{n}$ is either 0 or 2 .

Let $F$ denote the Cantor ternary set. Prove that $F+F=[0,2]$, if $F+F=\left\{ x+y: x \in F, y \in F\right\}$.

Let $F$ denote the Cantor ternary set. Prove that $F-F=[-1,1]$, if $F-F=\left\{ x-y: x \in F, y \in F\right\}$.

Prove or disprove: if $F$ is an uncountable dense subset of $\mathbb {R}$ then it is of second category.

Give examples of rational and irrational numbers that belong or do not belong to the Cantor ternary set.

Let $S$ denote the set of all real numbers whose base 3 representation requires using digit 1 . Prove that $S$ is dense in $\mathbb {R}$.

Let $\mathcal{F}$ be the collection of countable unions of intervals of $\mathbb {R}$. Prove that $\mathcal{F}$ is not closed under complementation.

Prove or disprove: every closed set in $\overline{\mathbb {R}}$ is a countable union of possibly degenerate closed intervals.

Prove that no countably infinite set can be perfect.

Prove Theorem 3.1.8.

The Fat Cantor set is constructed like the usual Cantor set but deleting smaller middle intervals at each step: middle quarters at step 1, middle sixteenths at step 2, and intervals of total length $1/4^{n}$ at step $n$, giving $F_{1} = [0, 3/8] \cup [5/8, 1]$, $F_{2} = [0, 5/32] \cup [7/32, 3/8] \cup [5/8, 25/32] \cup [27/32, 1]$, and so on; the Fat Cantor set is $F = \cap_{n=1}^{\infty } F_{n}$.

Theorem 3.1.8: The Fat Cantor set $F$ is a closed, uncountable, perfect, nowhere dense set whose every point is an accumulation point of $F^{C}$.

Let $F$ denote the Fat Cantor set obtained by deleting at step $n$ from each of the $2^{n-1}$ intervals the middle interval of length $\alpha^{n} I$, where $0<\alpha <1 / 2$. (In the classical Cantor set $\alpha =1 / 3$.) Prove that $F$ is a closed, uncountable, perfect, nowhere dense set whose every point is an accumulation point of $F^{C}$.

Prove that the characteristic function of a Fat Cantor Set $F$ is discontinuous at every point of $F$ and continuous at every point of $F^{C}$.

Prove Proposition 3.1.6.

Let $\left\{ O_{n}\right\}$ denote the deleted middle thirds in the construction of the Cantor set: $O_{1}=(1/3,2/3)$, $O_{2}=(1/9,2/9) \cup (1/3,2/3) \cup (7/9,8/9)$, and so on, with $O = \cup_{n=1}^{\infty } O_{n}$. Define a sequence of functions $\left\{ \varphi_{n}\right\}$ by: each $\varphi_{n}$ satisfies $\varphi_{n}(0)=0$, $\varphi_{n}(1)=1$; on the $2^{n}-1$ components of $O_{n}$ (in left-to-right order), $\varphi_{n}$ takes the constant values $1/2^{n}, 2/2^{n}, \ldots , (2^{n}-1)/2^{n}$ respectively; and $\varphi_{n}$ is extended linearly to the rest of $[0,1]$.

Proposition 3.1.6: The sequence $\left\{ \varphi_{n}\right\}$ defined above satisfies: (a) $\left|\varphi_{n+1}(x) - \varphi_{n}(x)\right| < 2^{-n}$ for every $x \in [0,1]$. (b) It converges uniformly on $[0,1]$. (The Cantor-Lebesgue function is $\varphi := \lim \varphi_{n}$.)

Prove Theorem 3.1.7.

Let $\left\{ O_{n}\right\}$ denote the deleted middle thirds in the construction of the Cantor set, $O = \cup_{n=1}^{\infty } O_{n}$, and let $\left\{ \varphi_{n}\right\}$ be the sequence of functions defined in Proposition 3.1.6 (each $\varphi_{n}(0)=0$, $\varphi_{n}(1)=1$, constant at the values $1/2^{n}, 2/2^{n}, \ldots , (2^{n}-1)/2^{n}$ on the components of $O_{n}$, extended linearly elsewhere). The Cantor-Lebesgue function is $\varphi := \lim \varphi_{n}$.

Theorem 3.1.7: The Cantor-Lebesgue function $\varphi$ is an increasing, continuous function that maps $[0,1]$ onto $[0,1]$. Its derivative exists on $O$ and $\varphi ' = 0$ on $O$.

Let $\varphi$ be the Cantor-Lebesgue function. Prove that $\varphi (1-x)=1-\varphi (x)$, for all $x \in [0,1]$.

Let $\varphi$ be the Cantor-Lebesgue function and let $0<\alpha \leq \log_{3} 2$ be fixed. Prove that there exists $C=C(\alpha )>0$ such that for all $x, y \in [0,1]$, $\left|\varphi (x)-\varphi (y)\right| \leq C\left|x-y\right|^{\alpha }$.

Prove that if $x=\sum_{n=1}^{\infty } a_{n} / 3^{n}$ is an element of the Cantor set, and $\varphi$ is the Cantor-Lebesgue function, then $\varphi (x)=\sum_{n=1}^{\infty } a_{n} / 2^{n+1}$.

Let $\varphi$ be the Cantor-Lebesgue function. Prove that $\varphi (x)=2 \varphi (x / 3)$, for all $x \in [0,1]$.

Let $F$ denote the Cantor set and $\varphi$ the Cantor-Lebesgue function. Prove or disprove: there is $c \in F \cap (0,1)$ for which $\varphi^{\prime }(c)$ exists and it is finite.

Find the length of the graph of Cantor-Lebesgue function.

Determine whether Cantor-Lebesgue function satisfies Lipschitz condition on $[0,1]$.

Let $f$ be an increasing function on $[0,1]$ that agrees with the CantorLebesgue function $\varphi$ on the complement of the Cantor set. Prove that $f=\varphi$ on $(0,1)$.