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Zariski topology (Definition)

Let $\A_k^n$ denote the affine space $k^n$ over a field $k$ . The Zariski topology on $\A_k^n$ is defined to be the topology whose closed sets are the sets $$ V(I) := \{ x \in \A_k^n \mid f(x) = 0 \text{ for all } f \in I\} \subset \A_k^n, $$ 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 $V \subset \A_k^n$ , the Zariski topology on $V$ is defined to be the subspace topology induced on $V$ as a subset of $\A_k^n$ .

Let $\P_k^n$ denote $n$ -dimensional projective space over $k$ . The Zariski topology on $\P_k^n$ is defined to be the topology whose closed sets are the sets $$ V(I) := \{ x \in \P_k^n \mid f(x) = 0 \text{ for all } f \in I\} \subset \P_k^n, $$ 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 $V \subset \P_k^n$ , the Zariski topology on $V$ is defined to be the subspace topology induced on $V$ as a subset of $\P_k^n$ .

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 $\A_k^1$ closed and regular morphisms continuous.




"Zariski topology" is owned by djao.
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Cross-references: closed, points, continuous, varieties, regular morphism, algebraic geometry, projective variety, homogeneous ideal, projective space, subset, induced, subspace topology, affine variety, polynomial ring, ideal, closed sets, topology, field, affine space
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This is version 1 of Zariski topology, born on 2002-05-11.
Object id is 2899, canonical name is ZariskiTopology.
Accessed 9610 times total.

Classification:
AMS MSC14A10 (Algebraic geometry :: Foundations :: Varieties and morphisms)

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