# Universal homeomorphism

In algebraic geometry, a universal homeomorphism is a morphism of schemes ${\displaystyle f:X\to Y}$ such that, for each morphism ${\displaystyle Y'\to Y}$, the base change ${\displaystyle X\times _{Y}Y'\to Y'}$ is a homeomorphism of topological spaces.

A morphism of schemes is a universal homeomorphism if and only if it is integral, radicial and surjective.[1] In particular, a morphism of locally of finite type is a universal homeomorphism if and only if it is finite, radicial and surjective.

For example, an absolute Frobenius morphism is a universal homeomorphism.

## References

1. EGA IV4, 18.12.11.
• Grothendieck, Alexandre; Dieudonné, Jean (1967). "Éléments de géométrie algébrique: IV. Étude locale des schémas et des morphismes de schémas, Quatrième partie". Publications Mathématiques de l'IHÉS. 32. doi:10.1007/bf02732123. MR 0238860.