Exponentiable maps and triquotients in Top

The concept of triquotients was introduced by Michael as a natural generalization of both open and perfect surjections. Niefield characterized exponentiable maps in Top by means of so called binding families of Scott-open sets in the fibers. Without recognizing this, Uspenskij used a characterization of triquotients by binding families of fiberwise nontrival Scott-open sets in order to show that arbitrary products of triquotients are again triquotients. This gives now rise to a proof of pullback-stability of triquotients in Top and a description of exponentiable maps as local triquotients of a special type. This sheds new light on exponentiable monomorphisms (injections). Furthermore, there is an elementary proof for Niefieldʹs characterization of exponentiable inclusions of subspaces as embeddings of locally closed subsets.