= Metric realization of a simplicial complex
{title2=$d_2(\lambda,\mu)=\left(\sum_v|\lambda_v-\mu_v|^2\right)^{1/2}$}
Represent a point by finitely supported <barycentric coordinates> $\lambda_v\geq0$ with $\sum_v\lambda_v=1$ and support a <simplex>. Restricting the displayed <Euclidean metric> on these coordinates gives a metric realization. The coordinate-sum is finite because both points have finite support. For a finite <simplicial complex>, this agrees with the <weak topology of a simplicial realization>: the coordinate map from the finite quotient realization is a continuous <bijection> from a <compact space> to a <Hausdorff space>, hence a <homeomorphism>. For an infinite complex the two topologies can differ, as the countable-star examples demonstrate.
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