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Product Metric

Given n metric spaces X_1,X_2,...,X_n, with metrics g_1,g_2,...,g_n respectively, the product metric g_1×g_2×...×g_n is a metric on the Cartesian product X_1×X_2×...×X_n defined as

(g_1×g_2×...×g_n)((x_1,x_2,...,x_n),(y_1,y_2,...,y_n))=sum_(i=1)^n1/(2^i)(g_i(x_i,y_i))/(1+g_i(x_i,y_i)).

This definition can be extended to the product of countably many metric spaces.

If for all i=1,...,n, X_i=R and g_i is the Euclidean metric of the real line, the product metric induces the Euclidean topology of the n-dimensional Euclidean space R^n. It does not coincide with the Euclidean metric of R^n, but it is equivalent to it.

See also

Productive Property

This entry contributed by Margherita Barile

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References

Cullen, H. F. Introduction to General Topology. Boston, MA: Heath, pp. 151-155, 1968.

Referenced on Wolfram|Alpha

Product Metric

Cite this as:

Barile, Margherita. "Product Metric." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/ProductMetric.html

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