Product Metric Spaces
If are metric spaces, and N is the Euclidean norm on Rn, then is a metric space, where the product metric is defined by
and the induced topology agrees with the product topology. By the equivalence of norms in finite dimensions, an equivalent metric is obtained if N is the taxicab norm, a p-norm, the max norm, or any other norm which is non-decreasing as the coordinates of a positive n-tuple increase (yielding the triangle inequality).
Similarly, a countable product of metric spaces can be obtained using the following metric
An uncountable product of metric spaces need not be metrizable. For example, is not first-countable and thus isn't metrizable.
Read more about this topic: Metric Space
Famous quotes containing the words product and/or spaces:
“In fast-moving, progress-conscious America, the consumer expects to be dizzied by progress. If he could completely understand advertising jargon he would be badly disappointed. The half-intelligibility which we expect, or even hope, to find in the latest product language personally reassures each of us that progress is being made: that the pace exceeds our ability to follow.”
—Daniel J. Boorstin (b. 1914)
“Deep down, the US, with its space, its technological refinement, its bluff good conscience, even in those spaces which it opens up for simulation, is the only remaining primitive society.”
—Jean Baudrillard (b. 1929)