Is there any theorem that says continuous injective function is an open map. I tried to scan few analysis books but could not find it. Can anyone suggest a reference for it?
Injective functions and Open mapping theorem.
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real-analysis
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5Without more detail about the domain and range, this is impossible to answer. For example, the map $f:\mathbb{R}\rightarrow\mathbb{R}^2$ given by $f(x) = (x,0)$ is about as a nice as possible (in particular, it's continuous and injective), but no where near open. It's image doesn't even contain an open set! – 2012-10-12
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0Here is a claim from one book that i found without any proof: – 2012-10-12
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2The closest thing I can think of now is the theorem *invariance of domain*. I don't know how to put a link here but you can just Wiki it. It says a continuous injective map is open if both the domain and codomain are Euclidean spaces of the same dimension. – 2012-10-12
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0Here is a claim from one book that i found without any proof:If $\Phi :D \rightarrow \Re^n $ is continuous and injective in the open subset of D of $\Re^n$, then $\Phi(D)$ is open. – 2012-10-12
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0http://math.stackexchange.com/questions/176526/open-mapping-of-the-unit-ball-into-itself/176527#176527 – 2012-10-12