A Short Introduction to Metric Spaces
Section 2: Subspaces and Product Spaces

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One of the things that we can do with metric spaces is to make "new spaces from old." For example, a subset of a metric space can be made into a metric space in its own right in a natural way. And given two metric spaces, their cross product can be made into a metric space; in fact, there are several reasonable ways to do that. This section looks at subspaces and products of metric spaces. The material on subspaces will be needed in later sections of this web site. The material on product spaces, although interesting, will not be needed and can be skipped.

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Subspaces

It should be clear that if $(M,d)$ is a metric space, then a subset of $M$ can be made into a metric space by using the same measure of distance on the subset as is used on $M.$ The subset with that inherited metric is called a "subspace."

The metric $d^{\prime }:X\times X\to \mathbb{R}$ is just the function $d$ restricted to the subset $X\times X$ of $M\times M.$ In practice, we will often use the same name for the distance function on a subspace as is used for the distance function on the containing space, but in this section, for clarity, I will avoid doing that.

A big part of understanding a metric space is understanding its open sets and closed sets. To understand a subspace $(X,d^{\prime })$ of $(M,d)$, we can start by noting that for $x\in X$, the open ball $B_r^{d^{\prime }}(x)$ in $(X,d^{\prime })$ is just the intersection with $X$ of the corresponding open ball in $(M,d)$. That is, $B_r^{d^{\prime }}(x)=B_r^d(x)\cap X$. This is clear because for $y\in X$, $d(x,y)=d^{\prime }(x,y)$ and therefore $d^{\prime }(x,y)<r$ if and only if $d(x,y)<r$. In fact, we get the following similar characterization for all open and closed sets in $(X,d^{\prime })$.

As an example, we can make the set of rational numbers, $\mathbb{Q}$, into a metric space $(\mathbb{Q},d^{\prime })$ using the subspace metric inherited from $(R,d)$. Let $A=\{x\in \mathbb{Q}|-\sqrt{2}<x<\sqrt{2}\}.$ Then $A$ is the intersection with $\mathbb{Q}$ of the open interval $(-\sqrt{2},\sqrt{2})$, which is an open set in $(\mathbb{R},d)$. By the theorem, $A$ is open in $(\mathbb{Q},d^{\prime }).$ But since $\sqrt{2}$ is not a rational number, $A$ is also a closed set in $(\mathbb{Q},d^{\prime })$ because $A=\mathbb{Q}\cap [-\sqrt{2},\sqrt{2}].$ There are many subsets of $\mathbb{Q}$ that are both open and closed in $(\mathbb{Q},d^{\prime })$. On the other hand, it can be shown that the only subsets of $\mathbb{R}$ that are both open an closed in $(\mathbb{R},d)$ are $\varnothing$ and $\mathbb{R}.$ (These facts will become significant have we have defined connected sets.)

It might also be instructive to note that $[1,2)$ is an open subset of the subspace $([1,3],d^{\prime })$ of $(\mathbb{R},d),$ even though it is not an open set in $(\mathbb{R},d).$ You are asked to verify this in one of the exercises in this section.

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Product Spaces

${\mathbb{R}}^2$ is the cross product $\mathbb{R}\times \mathbb{R}$. The usual metric on ${\mathbb{R}}^2$ can be defined in terms of the metric on $\mathbb{R}$. The usual distance, $\rho ,$ between the points $⟨x_1,y_1⟩$ and $⟨x_2,y_2⟩$ of ${\mathbb{R}}^2$ is given by $$\rho (⟨x_1,y_1⟩,⟨x_2,y_2⟩)=\sqrt{(x_1-x_2{)}^2+(y_1-y_2{)}^2}=\sqrt{d(x_1,x_2{)}^2+d(y_1,y_2{)}^2}.$$ This can be generalized to the cross product of any two metric spaces, $(M_1,d_1)$ and $(M_2,d_2)$. That is, we can define a metric space $(M_1\times M_2,\rho )$ where $$\rho (⟨x_1,y_1⟩,⟨x_2,y_2⟩)=\sqrt{d_1(x_1,x_2{)}^2+d_2(y_1,y_2{)}^2}$$ for any $x_1,x_2\in M_1$ and $y_1,y_2\in M_2.$ However, $\rho$ is not the only natural metric on $M_1\times M_2$, In particular, we can define two alternative metric spaces $(M_1\times M_2,\mu )$ and $(M_1\times M_2,\sigma )$ by $$\begin{array}{rl}\mu (⟨x_1,y_1⟩,⟨x_2,y_2⟩)& =max(d_1(x_1,x_2),d_2(y_1,y_2))\\ \sigma (⟨x_1,y_1⟩,⟨x_2,y_2⟩)& =d_1(x_1,x_2)+d_2(y_1,y_2)\end{array}$$

This illustration shows the open balls of radius $\epsilon$ about a point $a\in {\mathbb{R}}^2$ for each of the three product spaces $({\mathbb{R}}^2,\rho )$, $({\mathbb{R}}^2,\mu )$, and $({\mathbb{R}}^2,\sigma )$:

(However, I should note that all three metrics have the same open sets. That is, they produce the same topology on $M_1\times M_2$.)

We should prove that $\rho$, $\mu$, and $\sigma$ are in fact metrics on $M_1\times M_2$. I will leave $\mu$ and $\sigma$ as exercises, but here is a proof for $\rho .$

Of course, it is possible to define metrics on cross products of more than two metric spaces. For example, if $(M_1,d_1)$, $(M_2,d_2)$, $\dots ,(M_n,d_n)$ are metric spaces, we can define the metric $\rho$ on $M_1\times M_2\times \cdots \times M_n$ by $$\rho (⟨x_1,x_2,\dots x_n⟩,⟨y_1,y_2,\dots y_n⟩)=\sqrt{d_1(x_1,y_1{)}^2+d_2(x_2,y_2{)}^2+\cdots +d_n(x_n,y_n{)}^2}$$ The fact that this is a metric for all $n\ge 2$ can be proved by induction, noting that $$\begin{array}{rl}& \sqrt{d_1(x_1,y_1{)}^2+d_2(x_2,y_2{)}^2+\cdots +d_n(x_n,y_n{)}^2}\\ & =\sqrt{(\sqrt{d_1(x_1,y_1{)}^2+d_2(x_2,y_2{)}^2+\cdots +d_{n-1}(x_{n-1},y_{n-1}n{)}^2}{)}^2+d_n(x_n,y_n{)}^2}\end{array}$$ This definition, of course, gives us the usual metric on ${\mathbb{R}}^n.$ The metrics $\mu$ and $\sigma$ on $M_1\times M_2$ can similarly be extended to metrics on $M_1\times M_2\times \cdots \times M_n$.

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Exercises

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