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:
“Junk is the ideal product ... the ultimate merchandise. No sales talk necessary. The client will crawl through a sewer and beg to buy.”
—William Burroughs (b. 1914)
“Surely, we are provided with senses as well fitted to penetrate the spaces of the real, the substantial, the eternal, as these outward are to penetrate the material universe. Veias, Menu, Zoroaster, Socrates, Christ, Shakespeare, Swedenborg,—these are some of our astronomers.”
—Henry David Thoreau (1817–1862)