Editing Metric space (section)

===Product metric spaces===
{{main|Product metric}}
If <math>(M_1,d_1),\ldots,(M_n,d_n)</math> are metric spaces, and {{mvar|N}} is the [[Euclidean norm]] on <math>\mathbb R^n</math>, then <math>\bigl(M_1 \times \cdots \times M_n, d_\times\bigr)</math> is a metric space, where the [[product metric]] is defined by
<math display="block">d_\times\bigl((x_1,\ldots,x_n),(y_1,\ldots,y_n)\bigr) = N\bigl(d_1(x_1,y_1),\ldots,d_n(x_n,y_n)\bigr),</math>
and the induced topology agrees with the [[product topology]].  By the equivalence of norms in finite dimensions, a topologically equivalent metric is obtained if {{mvar|N}} is the [[taxicab norm]], a [[Norm (mathematics)#p-norm|p-norm]], the [[maximum norm]], or any other norm which is non-decreasing as the coordinates of a positive {{mvar|n}}-tuple increase (yielding the triangle inequality).

Similarly, a metric on the topological product of countably many metric spaces can be obtained using the metric
<math display="block">d(x,y)=\sum_{i=1}^\infty \frac1{2^i}\frac{d_i(x_i,y_i)}{1+d_i(x_i,y_i)}.</math>

The topological product of uncountably many metric spaces need not be metrizable. For example, an uncountable product of copies of <math>\mathbb{R}</math> is not [[first-countable space|first-countable]] and thus is not metrizable.