Zariski topology

Let $\mathbb{A}_{k}^{n}$ denote the affine space $k^{n}$ over a field $k$. The 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 $I\subset k[X_{1},\ldots,X_{n}]$ is any ideal in the polynomial ring $k[X_{1},\ldots,X_{n}]$. 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 $I\subset k[X_{0},\ldots,X_{n}]$ is any homogeneous ideal  in the graded $k$algebra $k[X_{0},\ldots,X_{n}]$. 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 ZariskiTopology 2013-03-22 12:38:11 2013-03-22 12:38:11 djao (24) djao (24) 4 djao (24) Definition msc 14A10 PrimeSpectrum