---
tag: H4FS63VY0Y
kind: definition
name: Topological field
msc: [12J99, 54H13]
defines: D7YQW9RG24
uses: [0QA8CHH8MG, 0CNQXVBHCX, 0P27W9RW0G]
sources:
  - url: https://en.wikipedia.org/wiki/Field_(mathematics)
---

## Definition

Let $K$ be a [[field|0QA8CHH8MG]] with a [[topology|0CNQXVBHCX]].
Give $K \times K$ the product topology and $K^\times = K \setminus \set{0}$ the subspace topology.
The field $K$ is a *topological field* if the following maps are [[continuous|0P27W9RW0G]]:

1. addition $K \times K \to K$, $(x, y) \mapsto x + y$;
2. negation $K \to K$, $x \mapsto -x$;
3. multiplication $K \times K \to K$, $(x, y) \mapsto xy$;
4. inversion $K^\times \to K^\times$, $x \mapsto x^{-1}$.

## Remarks

Clauses 1 to 3 say that $K$ is a topological ring.
Many authors also require the topology to be Hausdorff and not discrete.
Example: $\R$ with the usual topology is a topological field.
Clauses 1 to 3 are the limit rules for sums, negatives and products, and clause 4 holds because $x \mapsto 1/x$ is continuous on $\R \setminus \set{0}$.
More generally, every field with an absolute value, with the topology of the metric $d(x, y) = \abs{x - y}$, is a topological field.
Non-example: $\R$ with the lower limit topology, generated by the intervals $[a, b)$, is not a topological field.
Negation fails clause 2: the preimage of the open set $[0, 1)$ under $x \mapsto -x$ is $(-1, 0]$, which is not open.
Every field is a topological field for the discrete topology and for the indiscrete topology.