By Zorn leung
Does open mapping theorem holds in for n>1?
What is the target space? If . Trivially no. Example: .
For (say). Trivially yes.
The reason is, let be open and . Let’s say . Then it is easy to see that is nonconstant OR is non-constant (by Taylor’s expansion, say). Say is non-constant, then it maps a small (1-dim!) neighborhood of to an open set around , by 1-D open mapping theorem. So there is an open set around which is contained in .[My argument is wrong. ]
It turns out that I can correct my argument. Let my clarify it here.
For convenience, let’s assume be a non-constant holomorphic (entire) function. Let be open and . Wlog, we assume and
. We claim that there is an open set which contains . To see this, since is non-constant, there is with . Define to be an entire non-constant holomorphic function. Then by open mapping theorem, maps arbitrary small neighborhood of onto an open set around . So by choosing this neighborhood to be small enough, we see that there is an open set containing .
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