= Riemannian distance induces the manifold topology
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In a relatively <compact> coordinate ball the <Riemannian metric> norm is bounded above and below by positive multiples of the Euclidean norm. A coordinate segment gives the upper distance bound. A path leaving the ball has a positive lower length bound before its first exit; paths staying in it have the Euclidean displacement lower bound. Consequently distinct points have positive <Riemannian distance>, and the resulting <metric topology> equals the manifold topology. Connectedness makes the distance finite because <smooth manifolds> are locally path <connected>.
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