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 quasi-compact.

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