Continuous and Open maps

A function that is continuous and open is an embedding of a quotient of the original space. This is a very interesting notion, just like subquotients of groups. For instance, if you restrict a covering map to a subset of your domain, you (usually) get a continuous open map that is not one-to-one or surjective. This comes up a lot in geometry, for instance near cusps, or in creating the universal cover of graphs of groups; you look at the preimage of a subspace under a covering map (do it twice for two spaces with homeomorphic subspaces) and then glue together copies of the two spaces along these subspace... Anyways, I'm rambling, but such maps are interesting and useful and come up a lot, although withou any special name that I'm aware of.