# Zariski topology

Let ${\mathbb{A}}_{k}^{n}$ denote the affine space ${k}^{n}$ over a field $k$. The Zariski topology^{} on ${\mathbb{A}}_{k}^{n}$ is defined to be the topology whose closed sets are the sets

$$V(I):=\{x\in {\mathbb{A}}_{k}^{n}\mid f(x)=0\text{for all}f\in I\}\subset {\mathbb{A}}_{k}^{n},$$ |

where $I\subset k[{X}_{1},\mathrm{\dots},{X}_{n}]$ is any ideal in the polynomial ring $k[{X}_{1},\mathrm{\dots},{X}_{n}]$. For any affine variety^{} $V\subset {\mathbb{A}}_{k}^{n}$, the Zariski topology on $V$ is defined to be the subspace topology induced on $V$ as a subset of ${\mathbb{A}}_{k}^{n}$.

Let ${\mathbb{P}}_{k}^{n}$ denote $n$–dimensional projective space^{} over $k$. The Zariski topology on ${\mathbb{P}}_{k}^{n}$ is defined to be the topology whose closed sets are the sets

$$V(I):=\{x\in {\mathbb{P}}_{k}^{n}\mid f(x)=0\text{for all}f\in I\}\subset {\mathbb{P}}_{k}^{n},$$ |

where $I\subset k[{X}_{0},\mathrm{\dots},{X}_{n}]$ is any homogeneous ideal^{} in the graded $k$–algebra $k[{X}_{0},\mathrm{\dots},{X}_{n}]$. For any projective variety $V\subset {\mathbb{P}}_{k}^{n}$, the Zariski topology on $V$ is defined to be the subspace topology induced on $V$ as a subset of ${\mathbb{P}}_{k}^{n}$.

The Zariski topology is the predominant topology used in the study of algebraic geometry^{}. Every regular morphism of varieties^{} is continuous in the Zariski topology (but not every continuous map in the Zariski topology is a regular morphism). In fact, the Zariski topology is the weakest topology on varieties making points in ${\mathbb{A}}_{k}^{1}$ closed and regular morphisms continuous.

Title | Zariski topology |
---|---|

Canonical name | ZariskiTopology |

Date of creation | 2013-03-22 12:38:11 |

Last modified on | 2013-03-22 12:38:11 |

Owner | djao (24) |

Last modified by | djao (24) |

Numerical id | 4 |

Author | djao (24) |

Entry type | Definition |

Classification | msc 14A10 |

Related topic | PrimeSpectrum |