\documentclass[11
pt]{amsart}
%\usepackage{amssymb,amscd}\textheight=23cm

\usepackage{amsmath}
\usepackage{epstopdf}
\usepackage{amsthm}
\usepackage{times}

\usepackage{geometry}                % See geometry.pdf to learn the layout options. There are lots.
\geometry{letterpaper}                   % ... or a4paper or a5paper or ... 
%\geometry{landscape}                % Activate for for rotated page geometry
%\usepackage[parfill]{parskip}    % Activate to begin paragraphs with an empty line rather than an indent
\usepackage{graphicx}
%\usepackage{amssymb}
%\usepackage{epstopdf}
\DeclareGraphicsRule{.tif}{png}{.png}{`convert #1 `dirname #1`/`basename #1 .tif`.png}

\textwidth=15cm
%\titlepage
\topmargin=0pt
\headheight=0pt
\headsep=0pt
%\footskip=0pt
%\oddsidemargin=0pt
%\marginparsep=0pt
%\marginparwidth=0pt
\pagestyle{empty}

\newcommand{\R}{{\mathbb R}}
\newcommand{\C}{{\mathbb C}}
\newcommand{\Z}{{\mathbb Z}}
\newcommand{\Q}{{\mathbb Q}}
\newcommand{\N}{{\mathbb N}}

\newcommand{\T}{{\mathcal T}}
\newcommand{\B}{{\mathcal B}}
\newcommand{\X}{{\mathcal X}}

\newcommand{\vx}{{\mathbf x}}
\newcommand{\vy}{{\mathbf y}}
\newcommand{\vz}{{\mathbf z}}

\begin{document}
\parindent=0pt


{{\noindent Math 5510} \hfill {Homework 4--  Due  Wednesday 10/18/17}
\hfill {\ \ \ \ }\hfill}



\vskip.2cm

 \hrule



\bigskip

\begin{enumerate}
\item (2pts)  Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces.  
\begin{enumerate}
\item 
 Define a metric $d_{X\times Y}^\infty$ on their Cartesian product $X\times Y$ by
$$
d_{X\times Y}^\infty((x_1,y_1),(x_2,y_2)) = \max\{d_X(x_1,x_2),d_Y(y_1,y_2) \ | \ x_1,x_2\in X, y_1,y_2\in Y\}
$$
Prove that the topology defined by the metric $d_{X\times Y}^\infty$ is the same as the product topology of the metric topologies on $(X,d_X)$ and $(Y,d_Y)$.

\item
Prove that the same is true of the metric $d_{X\times Y}^1$ defined by
$$
d_{X\times Y}^1((x_1,y_1),(x_2,y_2)) =d_X(x_1,x_2)+d_Y(y_1,y_2).
$$
\emph{Suggestion} Prove that $d^\infty$ and $d^1$ are bi-Lipschitz.
\end{enumerate}
\vskip.3cm

\item (4 pts)  \emph{Metrics on infinite products}.    Let $\{X_i,d_i)_{i\in\N}$ be a countable collection of metric spaces, and suppose that the metrics $d_i$ are \emph{uniformly bounded}, more concretely, assume
$$
d_i(x,y)\le 1\ \hbox{ for all }\ i\in \N \ \hbox{ and for all }\ x,y\in X_i.
$$
Let $\mathcal{X} = \prod_{i=1}^\infty X_i$, the product of the $X_i$, with the product topology.  Denote elements of $\mathcal{X}$ by boldface  letters
$$
\vx = \{x_1,x_2,\dots\} =\{x_i\}, \vy = \{y_i \}, \ \hbox{ etc, }\ \   i \in\N, x_i, y_i \in X_i
$$
Define a metric $D$ on $\X$ by the formula
$$
D(\vx,\vy) = \sum_{i=1}^\infty \frac{d_i(x_i,y_i)}{2^i}
$$
\begin{enumerate}
  \item Prove that $D$ is a metric on $\X$ (check the three properties, in particular triangle inequality)
  \item Prove that for each $\vx\in\X$ and  for each $k\in\N$,  we  have the following two inclusions:
  \vskip.3cm
  $$
  \prod_{i=1}^{k+1} B_i(x_i,\frac{1}{2^{k+2}})\times \prod_{i=k+2}^\infty X_i \ \ \  \subset \  \  B_D(\vx,\frac{1}{2^k})\ \ \subset\ \   \prod_{i=1}^{k} B_i(x_i,\frac{1}{2^{k-i}})\times \prod_{i=k+1}^\infty X_i
  $$
  \vskip .5cm
 where $B_i(x_i,r_i)$ denotes the ball in $(X_i,d_i)$ of radius $r_i$  centered at $x_i$, the $i$th comoponent of $\vx$.
\vskip.3cm 
 \emph{Suggestion}: You need to write down the definition of the spaces involved and find the  inequalities that imply the containments.  For example, for the inclusion on the right, you have to use that $\vy\in B(\vx,\frac{1}{2^k})$ means
 $\sum_1^\infty \frac{d_i(x_i,y_i)}{2^i}\ \  < \frac{1}{2^k}$.  Show that this implies that $\vy\in \prod_1^\infty B_i(x_i,2^{i-k})$ .  \emph{Since the metrics are bounded by $1$}, that is, $d_i\le 1$ for all $i$, $B_i(x_i,2^{i-k} )= X_i$ for $i>k$. This would give the inclusion on the right.  Find a similar reasoning for the one on the left.
 \vskip.2cm
 \item \emph{Extra Credit} Prove that the metric topology of $D$ coincides with the product topology.
\end{enumerate}

\vskip .3cm

\item  (2 pts)  We saw in class the Cantor set $X=\{0,2\}^\N$ (same as $\prod_1^\infty \{0,2\})$  and the map $t:X\to  Y$, where $Y\subset [0,1]$ is the middle thirds Cantor set, given by
$$
t(\{a_i\}) = \sum_{i=1}^\infty \frac{a_i}{3^i}.
$$
We saw that this map is a continuous bijection from $X$ to $Y$.  Let us now prove that $t$ is Lipschitz.


Observe we are in the situation of the previous problem.  The metric spaces $X_i =\{0,2\}$ are \emph{bounded}.  The last problem asserts that $D$ gives the product topology. Let us use this assertion, with  a modification of $D$ to a metric which is  clearly equivalent, but better adapted to the Cantor set:
$$
D(\vx,\vy) = \sum_{i=1}^\infty \frac{d(x_i,y_i)}{3^i}
$$
Prove that
$$
|f(\vx) - f(\vy)| \le D(\vx, \vy)
$$

\vskip .3cm

\item (2 pts) Let $(X,\T_X)$ be a topological space, and give $X\times X$ the product topology.  Let $\Delta\subset X\times X$ be the \emph{diagonal}:
$$
\Delta =\{(x,y)\in X\times X\ |  \  x = y \} = \{ (x,x) \in X\times X \ | \ x\in X\}
$$
Prove that $X$ is a Hausdorff space if and only if $\Delta$ is a closed subset of $X\times X$ (in the product topology)

  \end{enumerate}  
   
 

                

 \end{document}
