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.
-
0http://math.stackexchange.com/questions/176526/open-mapping-of-the-unit-ball-into-itself/176527#176527 – 2012-10-12
2 Answers
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 $\mathbb{R} \times \{0\}$.
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.