A product space is meant to remember two kinds of motion at once. A point of $X \times Y$ has an $X$-coordinate and a $Y$-coordinate, so a path may move in the first coordinate, the second coordinate, or both. The first problem is not how to name the set $X \times Y$, but how to measure distance in it without losing either coordinate. If we measure only the first coordinate, different points with the same $X$-coordinate collapse. If we add the two coordinate distances, we get a usable answer, but it behaves differently from the Euclidean formula on $\mathbb R^2$. Product metrics are the systematic ways to combine coordinate distances into a genuine metric on a Cartesian product.

The guiding example is familiar: the plane is a product $\mathbb R \times \mathbb R$, and the usual Euclidean distance is built from the two coordinate distances by the Pythagorean rule. But topology is less rigid than geometry. For convergence, continuity, and compactness, many different formulas on $X \times Y$ give the same open sets. Product metrics separate two questions that are often blurred together: which formula measures distance, and which topology does that formula generate?

[example: A Bad Distance on a Product] Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces, and suppose $Y$ has at least two distinct points. Choose distinct points $y_1,y_2\in Y$, and choose any $x\in X$. Define \begin{align*} \rho((x_1,y_1),(x_2,y_2))=d_X(x_1,x_2). \end{align*} Then the two points $(x,y_1)$ and $(x,y_2)$ in $X\times Y$ are distinct because their second coordinates are distinct. However, substituting these points into the formula for $\rho$ gives \begin{align*} \rho((x,y_1),(x,y_2))=d_X(x,x). \end{align*} Since $d_X$ is a metric on $X$, the distance from a point to itself is $0$, so \begin{align*} d_X(x,x)=0. \end{align*} Therefore \begin{align*} \rho((x,y_1),(x,y_2))=0. \end{align*} Thus $\rho$ assigns distance $0$ to two distinct points of $X\times Y$, so it fails the identity of indiscernibles and is not a metric. The example shows that a product distance must detect motion in every coordinate. [/example]

The failure is small, but it is decisive. A metric is not only a numerical gadget; it determines which sequences converge, which functions are continuous, and which subsets are open. A product metric must keep coordinate information while still interacting well with the product structure.

## Definition

Before choosing a formula, we need the ambient object: a Cartesian product equipped with coordinate metric data. The construction begins with two metric spaces and asks for a distance between ordered pairs that responds to motion in both coordinates.

[definition: Product Metric] Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces. A product metric on $X\times Y$ is a function \begin{align*} D:(X\times Y)\times(X\times Y)&\to [0,\infty) \end{align*} which is a metric on $X\times Y$ and is defined from the coordinate metrics $d_X$ and $d_Y$ by a specified product metric formula. [/definition]

The associated metric space $(X\times Y,D)$ is often called a product metric space. The phrase is intentionally flexible: there are several standard formulas. The most important family is the $p$-family, which includes the Euclidean-style construction when $p=2$ and the taxicab-style construction when $p=1$.

## Standard Product Metric Formulas

For finite $p$, the idea is to measure the two coordinate distances and then apply the usual $\ell^p$ norm to the pair of nonnegative numbers. This imports the geometry of $\mathbb R^2$ into the abstract setting of arbitrary metric spaces.

[definition: $p$-Product Metric] Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces, and let $1 \le p < \infty$. The $p$-product metric on $X \times Y$ is the function $d_p:(X\times Y)\times(X\times Y)\to [0,\infty)$ defined by \begin{align*} d_p((x_1,y_1),(x_2,y_2))=\bigl(d_X(x_1,x_2)^p+d_Y(y_1,y_2)^p\bigr)^{1/p}. \end{align*} [/definition]

The finite $p$-formula is excellent for estimates that combine coordinate errors by summing powers. It still leaves a useful limiting question: what if we care only about the worst coordinate error, so that a product ball forces both coordinates to stay inside prescribed balls of the same radius?

[definition: Supremum Product Metric] Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces. The supremum product metric on $X \times Y$ is the function $d_\infty:(X\times Y)\times(X\times Y)\to [0,\infty)$ defined by \begin{align*} d_\infty((x_1,y_1),(x_2,y_2))=\max\{d_X(x_1,x_2),d_Y(y_1,y_2)\}. \end{align*} [/definition]

Before these formulas can support convergence or continuity arguments, they must pass the metric axioms. Positivity and symmetry are inherited from the coordinate metrics, but the main question is whether combining coordinate distances by a $p$-formula still produces a valid triangle inequality on the product.

[quotetheorem:8620]

This theorem justifies the terminology. From this point onward, the formulas are not candidates or analogies; they are genuine distances on the product. The next issue is how these distances compare with each other, since different choices of $p$ measure the same coordinate errors with different emphasis.

[example: Familiar Product Metrics on $\mathbb R^2$] Take $X=Y=\mathbb R$ with the usual metric $d(s,t)=|s-t|$, and let $a=(0,0)$ and $b=(3,4)$. The coordinate distances are \begin{align*} d(0,3)=|0-3|=|-3|=3. \end{align*} and \begin{align*} d(0,4)=|0-4|=|-4|=4. \end{align*} For the taxicab product metric, \begin{align*} d_1(a,b)=d(0,3)+d(0,4)=3+4=7. \end{align*} For the Euclidean product metric, \begin{align*} d_2(a,b)=\bigl(d(0,3)^2+d(0,4)^2\bigr)^{1/2}=(3^2+4^2)^{1/2}. \end{align*} Since $3^2=9$ and $4^2=16$, this becomes \begin{align*} d_2(a,b)=(9+16)^{1/2}=25^{1/2}=5. \end{align*} For the supremum product metric, \begin{align*} d_\infty(a,b)=\max\{d(0,3),d(0,4)\}=\max\{3,4\}=4. \end{align*} Thus the same pair of points has distances $7$, $5$, and $4$ under $d_1$, $d_2$, and $d_\infty$, respectively. Both coordinate distances are nonzero, so all three metrics detect motion in both coordinates; by *[Product Metrics Induce the Product Topology](/theorems/8622)*, all three generate the usual topology on $\mathbb R^2$. [/example]

## Coordinate Control and Convergence

### Comparing Product Metrics

A product metric is useful because it translates statements about ordered pairs into statements about coordinates. This is the main reason product metrics appear throughout analysis: convergence in a product should mean convergence in each coordinate, and continuity into or out of a product should be testable coordinate by coordinate.