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)^n 1/2^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.
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