---
tag: 6RF8CB69KQ
kind: concept
name: Linear topology
msc: [13J99, 16W80]
sources:
  - url: https://en.wikipedia.org/wiki/Linear_topology
same_as:
  wikidata: Q6553531
  wikipedia: Linear topology
---

## Summary

A linear topology on a ring or a module is a topology in which the neighbourhoods of $0$ are generated by [[ideals|096Z9MP2ME]] or [[submodules|05WDSN597C]].
It makes the ring a topological ring whose small neighbourhoods of $0$ are closed under addition and under multiplication by ring elements.
The main examples are the [[I-adic topologies|2VF2TSJ3PY]], which lead to [[completions|KBN13NPYJQ]] such as the $p$-adic integers and formal power series rings.

## Explanations

### L1 human

In a linear topology, "being close to $0$" means "lying in a small ideal".
Two elements are close when their difference lies in a small ideal from a chosen family.
For the integers with the $p$-adic topology, the chosen ideals are $p^n \Z$.
So $p^{10}$ is closer to $0$ than $p$, and the numbers $1, 1 + p, 1 + p + p^2, \ldots$ get closer and closer to each other.