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:
“The seashore is a sort of neutral ground, a most advantageous point from which to contemplate this world. It is even a trivial place. The waves forever rolling to the land are too far-traveled and untamable to be familiar. Creeping along the endless beach amid the sun-squall and the foam, it occurs to us that we, too, are the product of sea-slime.”
—Henry David Thoreau (1817–1862)
“When I consider the short duration of my life, swallowed up in the eternity before and after, the little space which I fill and even can see, engulfed in the infinite immensity of spaces of which I am ignorant and which know me not, I am frightened and am astonished at being here rather than there. For there is no reason why here rather than there, why now rather than then.”
—Blaise Pascal (1623–1662)