# Continuous and Open maps

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.

Thanks!

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A continuous function that maps open sets to open sets is just called an open map as far as I know. – echoone Jun 1 '12 at 23:09
It would seem to me that the most important thing about a continuous open map is that it will be a quotient map onto its image. – Iian Smythe Jun 1 '12 at 23:21
@ismythe: That's very true, but I guess that falls under the surjective case. – Francis Adams Jun 1 '12 at 23:31
I think for some authors (the Wikipedia page gives this impression, for example) "open" maps are not required to be continuous. So just be careful. – Dylan Moreland Jun 1 '12 at 23:41
@echoone: I’ve certainly never assumed that open maps are continuous. In particular, it’s a commonplace that the inverse of a continuous bijection is an open bijection, but not necessarily continuous. – Brian M. Scott Jun 2 '12 at 6:06