Definition
Category of topological fields
Hausdorff topological fields and continuous field embeddings.
The category of topological fields used here has fields with a Hausdorff topology making addition, multiplication, additive inversion, and inversion on the nonzero elements continuous. Morphisms are continuous unit-preserving field homomorphisms. Thus every morphism is injective, but is not required to be a topological embedding onto its image by definition.
Convention and comparison
The category includes archimedean fields and nonarchimedean local fields with their specified topologies. A field with the discrete topology is a different topological-field view from the same field with a nondiscrete topology.