---
tag: D7YQW9RG24
kind: concept
name: Topological field
msc: [12J99, 54H13]
specializes: [04C5ZBG1WT]
sources:
  - url: https://en.wikipedia.org/wiki/Field_(mathematics)
---

## Summary

A topological field is a [[field|04C5ZBG1WT]] with a [[topology|0CQHFPGJ45]] in which addition, multiplication, negation and inversion are continuous, see the [[definition|H4FS63VY0Y]].
It is a field that is also a topological ring and in which taking inverses is continuous.
The real numbers, the complex numbers, the [[p-adic numbers|BT6T702C28]] and every field with an absolute value are topological fields.
The notion is the common setting for analysis over fields other than $\R$ and $\C$.

## Explanations

### L1 human

In $\R$, small changes of $x$ and $y$ give small changes of $x + y$, $xy$ and, when $x \ne 0$, of $1/x$.
A topological field is any field with a notion of "closeness" for which the same is true.
The $p$-adic numbers are an example where two numbers are close when their difference is divisible by a high power of $p$.
The rule that $1/x$ moves little when $x$ moves little, away from $0$, is the extra demand compared with a topological ring.