The natural pseudo-distance is a similarity measure conceived for the purpose of comparing shapes. In this paper we revisit this pseudo-distance from the point of view of quotients. In particular, we show that the natural pseudo-distance coincides with the quotient pseudo-metric on the space of continuous functions on a compact manifold, endowed with the uniform convergence metric, modulo self-homeomorphisms of the manifold. As applications of this result, the natural pseudo-distance is shown to be actually a metric on a number of function subspaces such as
the space of topological embeddings, of isometries, and of simple Morse functions on surfaces
