---
tag: N6JX9BCKQP
kind: concept
name: Basic open set
msc: [14A05, 13A15]
sources:
  - url: https://en.wikipedia.org/wiki/Spectrum_of_a_ring
---

## Summary

A basic open set of the [[spectrum|FB45BHHA2J]] of a [[commutative ring|0310REJJY8]] $R$ is a set $D(f)$ of [[prime ideals|00MHHZ9SNE]] not containing a fixed element $f$.
These sets form a [[basis|0NA0732YAG]] of the [[Zariski topology|8SQK7RTB63]].
The set $D(f)$ is in natural bijection with the spectrum of the [[localization|4C1AE84N5N]] $R_f$, so basic open sets are again spectra of rings.

## Explanations

### L1 human

Think of $f$ as a function on the space of prime ideals.
The basic open set $D(f)$ is the set of points where $f$ does not vanish.
On $\operatorname{Spec} \Z$, the set $D(6)$ consists of all points except $(2)$ and $(3)$.
Every open set is a union of such sets, just as every open set of the plane is a union of discs.
The [[definition|PS8DVS9WTA]] states this.