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Concept# Fréchet surface

Summary

In mathematics, a Fréchet surface is an equivalence class of parametrized surfaces in a metric space. In other words, a Fréchet surface is a way of thinking about surfaces independently of how they are "written down" (parametrized). The concept is named after the French mathematician Maurice Fréchet.
Definitions
Let M be a compact 2-dimensional manifold, either closed or with boundary, and let (X, d) be a metric space. A parametrized surface in X is a map
f : M \to X
that is continuous with respect to the topology on M and the metric topology on X. Let
\rho(f, g) = \inf_{\sigma} \max_{x \in M} d(f(x), g(\sigma(x))),
where the infimum is taken over all homeomorphisms \sigma of M to itself. Call two parametrized surfaces f and f in X equivalent if and only if
\rho(f, g) = 0.

Official source

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