Section Products of Metric Spaces
If we have two metric spaces \((X, d_1)\) and \((X_2 ,d_2)\text{,}\) we might wonder if we can make the set \(X_1 \times X_2\) into a metric space. A natural approach might be to define a function \(d : (X_1 \times X_2) \times (X_1 \times X_2) \to \R\) by
\begin{equation*}
d((x,y),(u,v)) = d_1(x,u)d_2(y,v)
\end{equation*}
for \((x,y)\) and \((u,v)\) in \(X_1 \times X_2\text{.}\) However, this \(d\) does not define a metric. For example, if \(x \in X_1\) and \(y \neq v\) in \(X_2\text{,}\) then \(d((x,y),(x,v)) = 0\) even though \((x,y) \neq (x,v)\text{.}\) To make a metric, we can take a clue from the Euclidean metric on \(\R \times \R\text{.}\) On \(\R\text{,}\) the metric has the form \(d_1(x,y) = |x-y|\text{,}\) while on \(\R^2\) the metric is
\begin{align*}
d_E((x_1,x_2), (y_1,y_2)) \amp = \sqrt{ (x_1-y_1)^2 + (x_2-y_2)^2}\\
\amp = \sqrt{d_1(x_1,y_1)^2 + d_1(x_2,y_2)^2}\text{.}
\end{align*}
So on \((X, d_1)\) and \((X_2 ,d_2)\) we could consider defining \(d : (X_1 \times X_2) \times (X_1 \times X_2) \to \R\) by
\begin{equation}
d((x_1,x_2), (y_1,y_2)) = \sqrt{ d_1(x_1,y_1)^2 + d_2(x_2,y_2)^2}\text{.}\tag{11.1}
\end{equation}
Activity 11.3.
In this activity we verify some of the properties that make
\(d\) as defined in
(11.1) a metric. Let
\(x=(x_1,x_2)\) and
\(y=(y_1,y_2)\) be in
\(X_1 \times X_2\)
(a)
Explain why \(d(x,y)\) is greater than or equal to \(0\text{.}\)
(b)
Explain why \(d(x,y) = d(y,x)\text{.}\)
(c)
Explain why \(d(x,y) = 0\) if and only if \(x = y\text{.}\)
Activity 11.3 provides three of the four items that are necessary to prove that
\(d\) as defined in
(11.1) is a metric. We verify the last property, the triangle inequality, now.
Let
\(x\) and
\(y\) be defined as in
Activity 11.3, and let
\(z = (z_1, z_2)\) be in
\(X_1 \times X_2\text{.}\) Then
\begin{align*}
d(x,z)^2 \amp = d_1(x_1,z_1)^2 + d_2(x_2,z_2)^2 \\
\amp \leq \left(d_1(x_1,y_1) + d_1(y_1,z_1)\right)^2 + \left(d_2(x_2,y_2) + d_2(y_2,z_2)\right)^2 \\
\amp = \left(d_1(x_1,y_1)^2 + d_2(x_2,y_2)^2\right) + 2\left(d_1(x_1,y_1)d_1(y_1,z_1) + d_2(x_2,y_2)d_2(y_2,z_2) \right) \\
\amp \qquad + \left(d_1(y_1,z_1)^2 + d_2(y_2,z_2)^2\right)\\
\amp = d(x,y)^2 + 2\left(d_1(x_1,y_1)d_1(y_1,z_1) + d_2(x_2,y_2)d_2(y_2,z_2)\right) + d(y,z)^2 \\
\amp \leq d(x,y)^2 + d(y,z)^2\text{.}
\end{align*}
Since all terms are non-negative we conclude that
\begin{align*}
d(x,z) \leq \sqrt{d(x,y)^2 + d(y,z)^2} \amp \leq \sqrt{d(x,y)^2 + 2 d(x,y)d(y,z) + d(y,z)^2} \\
\amp = \sqrt{\left(d(x,y)+d(y,z)\right)^2} \\
\amp = d(x,y) + d(y,z)\text{.}
\end{align*}
We conclude that
\(d\) as defined in
(11.1) is a metric on
\(X_1 \times X_2\text{,}\) and so we can make the product of any two metric spaces into a metric space.
In the next activity we consider products of open balls and open sets in products of metric spaces.
Activity 11.4.
Let \(X_1 = [1,2]\) and \(X_2 = [3,4]\) as subspaces of \(\R^2\) using the Euclidean metric.
(a)
Explain in detail what the product space \(X_1 \times X_2\) looks like.
(b)
If \(B_1\) is an open ball in \(X_1\) and \(B_2\) is an open ball in \(X_2\text{,}\) is \(B_1 \times B_2\) an open ball in \(X_1 \times X_2\text{?}\) Explain.
(c)
If \(B_1\) is an open ball in \(X_1\) and \(B_2\) is an open ball in \(X_2\text{,}\) is \(B_1 \times B_2\) an open set in \(X_1 \times X_2\text{?}\) Explain.
(d)
Find, if possible, an open subset of \(X_1 \times X_2\) that is not of the form \(O_1 \times O_2\) where \(O_1\) is open in \(X_1\) and \(O_2\) is open in \(X_2\text{.}\)
Activity 11.4 shows that open sets in a product are more complicated than just products of open sets in the factors. We will return to product later when we consider topological spaces.
We conclude with one final comment about products. We can make the Cartesian product of any number of metric spaces into a metric space with the same construction we used for the product of two spaces.
Definition 11.4.
Let \((X_i, d_i)\) be metric spaces for \(i\) from \(1\) to some positive integer \(n\text{.}\) The product metric space \((X,d)\) is the Cartesian product
\begin{equation*}
X = X_1 \times X_2 \times \cdots \times X_n = \prod_{i=1}^n X_i
\end{equation*}
with metric \(d\) defined by
\begin{equation*}
d(x,y) = \sqrt{\sum_{i=1}^n d_i(x_i,y_i)^2}
\end{equation*}
when \(x = (x_1, x_2, \ldots,
x_n)\) and \(y = (y_1, y_2, \ldots, y_n)\) are in \(X\text{.}\)
The metric
\(d\) is called the
product metric and the spaces
\((X_i,d_i)\) are called the
coordinate or
factor spaces of
\((X,d)\text{.}\) The proof that
\(d\) is a metric is essentially the same as in the
\(n=2\) case, and is left to
Exercise 6.
