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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?

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    Without 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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    Here is a claim from one book that i found without any proof:2012-10-12
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    The 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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    Here 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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    http://math.stackexchange.com/questions/176526/open-mapping-of-the-unit-ball-into-itself/176527#1765272012-10-12

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Without more assumptions about the domain and codomain of the function, this is not necessarily true. As a particular example, the function $$f : \mathbb{R} \to \mathbb{R}^2, f(x) = (x, 0)$$ is a continuous function which is clearly injective. However, $f$ is not an open map: Any open subset of $\mathbb{R}^2$ cannot be contained within $\mathbb{R} \times \{0\}$.

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HuiYu: Just paste the link in the comment: http://en.wikipedia.org/wiki/Invariance_of_domain

user42574: The relevant theorem is the invariance of domain; see the link. You won't see this proved in most analysis books, since this is a statement of topology and typically is proved using the Brouwer fixed-point theorem. You may find the invariance of domain proved in Topology by J. Munkres, a fairly standard reference on topology.