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Product Metrics


Given $n$ sets each with metrics, there is a standard way of turning the direct product of the sets into a metric space. In other words, defining a distance on the tuples of elements from the sets.

Motivating result

Let $(A_1, d_1), \dots , (A_n, d_n)$ be metric spaces. Let $A$ be $\prod_{i = 1}^n A_n$ and let $R$ be the set of real numbers. Define $d: A \times A \to R$ by

\[ d(a, b) = \max\set{d_1(a_1, b_1), \dots , d_n(a_n, b_n)}. \]

Then $(A, d)$ is a metric space.

We call $d$ the product metric.
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