Zariski topology

2020 Mathematics Subject Classification: Primary: 14-XX [MSN][ZBL]

The Zariski topology on an affine space $A^n$ is the topology defined on $A^n$ by taking the closed sets to be the algebraic subvarieties of $A^n$. If $X$ is an affine algebraic variety (see Affine algebraic set) in $A^n$, the induced topology on $X$ is also known as the Zariski topology. In a similar manner one defines the Zariski topology of the affine scheme ${\rm Spec}\; A$ of a ring $A$ (sometimes called the spectral topology) — the closed sets are all the sets $$V(\mathfrak l) = \{{\mathfrak p}\in {\rm Spec A} \mid {\mathfrak p} \supset {\mathfrak l}\},$$ where ${\mathfrak l}$ is an ideal of $A$.

The Zariski topology was first introduced by O. Zariski [Za], as a topology on the set of valuations of an algebraic function field. Though, in general, the Zariski topology is not separable, many constructions of algebraic topology carry over to it [Se]. An affine scheme endowed with the Zariski topology is a quasi-compact space.

The topology most naturally defined on an arbitrary scheme is also called the Zariski topology in order to distinguish between it and the étale topology, or, if the variety $X$ is defined over the field ${\mathbb C}$, between it and the topology of an analytic space on the set of complex-valued points of $X({\mathbb C})$.

References

[Ha]	R. Hartshorne, "Algebraic geometry", Springer (1977) pp. Sect. IV.2 MR0463157 Zbl 0367.14001
[Se]	J.-P. Serre, Fibre spaces and their applications, Moscow (1958) pp. 372–450 (In Russian; translated from French)
[Za]	O. Zariski, "The compactness of the Riemann manifold of an abstract field of algebraic functions" Bull. Amer. Math. Soc., 50 : 10 (1944) pp. 683–691 MR0011573 Zbl 0063.08390