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.*



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> Are you aware of other results in the same spirit?
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> Is there a general framework that somehow encompasses all these results?
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Injectivity implies surjectivity
A famous result in this spirit is the [Ax-Grothendieck theorem](https://en.wikipedia.org/wiki/Ax%E2%80%93Grothendieck_theorem), whose statement is the following:



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> **Theorem.** If $f \colon \mathbb{C}^n \to \mathbb{C}^n$ is an injective polynomial function then $f$ is bijective.
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