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.


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    $\begingroup$ A continuous function that maps open sets to open sets is just called an open map as far as I know. $\endgroup$
    – echoone
    Jun 1 '12 at 23:09
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    $\begingroup$ 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. $\endgroup$ Jun 1 '12 at 23:21
  • $\begingroup$ @ismythe: That's very true, but I guess that falls under the surjective case. $\endgroup$ Jun 1 '12 at 23:31
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    $\begingroup$ 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. $\endgroup$ Jun 1 '12 at 23:41
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    $\begingroup$ @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. $\endgroup$ Jun 2 '12 at 6:06

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.


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