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We give a version in characteristic $p>0$ of Mumford's theorem characterizinga smooth complex germ of surface $(X,x)$ by the triviality of the topologicalfundamental group of $U=X\setminus \{x\}$.  This note relies on discussions the authors had during the Christmas break2009/10 in Ivry. They have been written down by H\'el\`ene in the night whenEckart died, as a despaired sign of love.

Surface singularities dominated by smooth varieties