= Weighted binary-sequence metric
{title2=$d(x,y)=\sum_{n\geq1}2^{-n}|x_n-y_n|$}
On $\{0,1\}^{\mathbb N}$ this sum is a <metric>, since each summand obeys the <triangle inequality> and a zero sum forces coordinatewise equality. A prefix <cylinder set> fixing the first $n$ coordinates contains the <open ball> of radius $2^{-n}$ about each of its points and has <diameter> at most $2^{-n}$. These bounds show that the <metric topology> equals the prefix <cylinder set> topology, the usual <Cantor space> topology.
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