---
tag: TSF8RZDD5B
kind: concept
name: Topological ring
msc: [16W80, 54H13]
specializes: [0B1261B5ED]
sources:
  - url: https://en.wikipedia.org/wiki/Topological_ring
same_as:
  wikidata: Q922339
  wikipedia: Topological ring
---

## Summary

A topological ring is a [[ring|0B1261B5ED]] with a [[topology|0CQHFPGJ45]] in which addition, negation and multiplication are continuous.
It lets one speak of limits, convergent series and completions inside a ring.
The real and complex numbers, the [[$p$-adic numbers|BT6T702C28]] and rings of formal power series with the $(x)$-adic topology are standard examples.
A [[topological field|D7YQW9RG24]] is a topological ring that is a field and in which inversion is also continuous on nonzero elements.

## Explanations

### L1 human

A topological ring is a ring where it makes sense to say that elements are "close" to each other, and the ring operations respect closeness.
If $a$ is close to $a'$ and $b$ is close to $b'$, then $a + b$ is close to $a' + b'$ and $ab$ is close to $a'b'$.
The real numbers with the usual distance are the basic example.
In the integers with the $p$-adic topology, a number is close to $0$ when it is divisible by a high power of $p$.