nearfaithful

nearfaithful is a term used in mathematics, particularly in the study of dynamical systems and topology, to describe a specific type of approximation between two functions or maps. A map f from a topological space X to itself is said to be nearfaithful if for every non-empty open set U in X, the image of U under f, denoted f(U), is also a non-empty open set, and furthermore, the preimage of f(U), denoted f⁻¹(f(U)), is not "too much larger" than U. More precisely, there exists a constant C such that for every non-empty open set U, the measure of f⁻¹(f(U)) is less than or equal to C times the measure of U. The precise definition can vary depending on the context and the specific measure being used.

This concept is a relaxation of the property of being a homeomorphism or a covering map. A homeomorphism is a map that preserves topological structure and is bijective. A covering map is surjective and has the property that every point in the target space has a neighborhood whose preimage consists of disjoint open sets, each mapping homeomorphically onto the neighborhood. Nearfaithfulness allows for some "loss of information" or "compression" in the mapping, but still requires that open sets are mapped to open sets and that the preimages of these images are not excessively large. It plays a role in understanding how maps distort or preserve local structure and is relevant in areas like ergodic theory and geometric measure theory.