Jumpstart - Homework Assignment 2

Problem 1

For $x, y\in \mathbb{R}$ define \begin{equation*} d_1(x,y)=(x-y)^2\, ,\: d_2(x,y)=\sqrt{|x-y|},\: d_3(x,y)=|x^2-y^2|,\: d_4(x,y)=|x-2y|,\text{ and } d_5(x,y)=\frac{|x-y|}{1+|x-y|}. \end{equation*} Determine, for each of these, whether it is a metric or not.

Problem 2

Let $X=C(M)$ be the set of all continuous real-valued functions on a compact metric space $M$. Define $d: X\times X\to [0,\infty)$ by $$ d(f, g)=\max\{|f(x)-g(x)|: x\in M\} $$ Prove that $d$ is a well-defined metric on $C(M)$.

Problem 3

If a given metric space $(X,d)$ were not complete, how would you go about constructing its completion?

Problem 4

Let $(X,\|\cdot\|)$ be a complete normed vector space, i.e. a vector space endowed with a norm in which Cauchy sequences converge. Prove that $$ \sum _{n\in \mathbb{N}}\| x_n\|<\infty\Longrightarrow\:\exists\: x\in X\text{ s.t. }\|x-\sum _{n=1}^N x_n\|\to 0\text{ as }N\to\infty\, . $$

Problem 5

Let $(X,d_0)$ be a set endowed with the discrete metric. Find all balls $B(x, r)$ in $(X, d_0)$.

Problem 6

Let $E$ be a nonempty subset of a metric space $X$ and define the distance from $x\in X$ to $E$ by $\rho_E(x)=\inf\{ d(x,y)\, |\, y\in E\}$. Prove that
(a) $\rho_E(x)=0$ if and only if $x\in \overline{E}$.
(b) $\rho_E$ is uniformly continuous function on $X$ by showing that $$ |\rho_E(x)-\rho_E(y)|\le d(x,y),\: x, y\in X. $$

Problem 7

Construct a set of real numbers which has exactly three limit points.

Problem 8

Let $(X,d)$ be a metric space. Prove or disprove:
(a) The intersection of finitely many dense subsets of $X$ is dense in $X$.
(b) The intersection of finitely many open dense subsets of $X$ is open and dense.

Problem 9

Let $\mathbb{R}[x]$ be the vector space of all polynomials with real coefficients. For $p=\sum _{k=0}^n a_kx^k\in \mathbb{R}[x]$ define its norm by $$ \| p\| =\max_{k=0,\dots,n}|a_k|. $$ Show that $\bigl( \mathbb{R}[x],\|\cdot\|\bigr)$ is not complete.

Problem 10

Let $X$ be an arbitrary set. A subset $\mathcal{O}$ of the power set $2^X$ of $X$ is called a topology on $X$ if it satisfies the following conditions:
(t1) $X,\emptyset\in \mathcal{O}$.
(t2) If $O_\alpha\in \mathcal{O}$ for $\alpha\in A$ and arbitrary index set $A$, then $\bigcup _{\alpha\in A}O_\alpha\in \mathcal{O}$.
(t3) If $O_k\in \mathcal{O}$ for $k=1,\dots,n$ and arbitrary $n\in \mathbb{N}$, then $\bigcap _{k=1}^nO_k\in \mathcal{O}$.
Let $(X,\mathcal{O})$ be a topological space (i.e. a set with a topology) and let $Y\subset X$ be an arbitrary subset. Prove that $$ \mathcal{O}_Y:=\{ O\cap Y\, :\, O\in \mathcal{O}\} $$ is a topology on $Y$. It is called the induced or relative topology of $Y$. Its elements are said to be relatively open in $Y$. If $X=\mathbb{R}$ and $Y=[0,1)$, describe of all relatively open sets in $Y$.