I was reading through Munkres' Topology and in the section on Continuous Functions, these three statements came up:
If a function is continuous, open, and bijective, it is a homeomorphism.
If a function is continuous, open, and injective, it is an imbedding.
If a function is continuous, open, and surjective, it is a quotient map. (This one isn't a definition, but it is a particular example.)
So then I wondered: is there was a name for functions that are just continuous and open without being 1-1 or onto? Are these special at all? Or does dropping the set theoretic restrictions give us a class of functions that just isn't very nice.
EDIT: This question is not asking if continuous implies open or vice versa. I know we can have one of them, both, or neither. The question is about if we suppose we have both of them, but our function isn't 1-1 or onto, what can we say about this function.