TITLE: Injectivity implies surjectivity
QUESTION [85 upvotes]: In some circumstances, an injective (one-to-one) map is automatically surjective (onto). For example,
Set theory
An injective map between two finite sets with the same cardinality is surjective.
Linear algebra
An injective linear map between two finite dimensional vector spaces of the same dimension is surjective.
General topology
An injective continuous map between two finite dimensional connected compact manifolds of the same dimension is surjective.

Are you aware of other results in the same spirit?
Is there a general framework that somehow encompasses all these results?

REPLY [2 votes]: There is an example of this in stable homotopy theory, related to the famous `Generating Hypothesis,' conjectured by Freyd (and still open as far as I know!).   For finite complexes $X$ and $Y$, the map from the stable homotopy classes of (pointed) maps from $X$ to $Y$ can be related to the algebraic
maps between stable homotopy groups via
$$
[ X, Y]^S \to \mathrm{Hom}(\pi_*^S(X), \pi_*^S(Y) ).
$$
The Generating Hypothesis says that this function is always injective
finite complexes $X$ and $Y$;  and it is a theorem of Freyd that if 
it is true, then the map is always surjective, too.
It has been long enough since I read up on this that I can't remember if the surjectivity follows on a pointwise basis, or whether you need the full conjecture to get the implication for a particular $X$ and $Y$.