The natural pseudo-distance as a quotient pseudo-metric, and applications

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

Injectivity and sections

We investigate injectivity in a comma-category C/B using the notion of the "object of sections" S(f) of a given morphism f:X--> B in C. We first obtain that f:X--> B is injective in C/B if and only if the morphism :X--> XxB is a section in C/B and the object S(f) of sections of f is injective in C. Using this approach, we study injective objects f with respect to the class of embeddings in the categories ContL/B (AlgL/B) of continuous (algebraic) lattices over B. As a result, we obtain both topological (every fiber of f has maximum and minimum elements and f is open and closed) and algebraic (f is a complete lattice homomorphism) characterizations