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This is a topology, because two basic axioms can be checked. The intersection of any number of closed sets is the set of mutual zeros of the union of all the defining polynomials. The union of two closed sets, defined by polynomials Pi and Qj, is the set of mutual zeros of the set of products PiQj.
This definition indicates the kind of space that can be given a Zariski topology: for example, we define the Zariski topology on an n-dimensional vector space Fn over a field F, using the definition above. That this definition yields a true topology is easily verified.
The Zariski topology given to some finite-dimensional vector space doesn't depend on the specific basis chosen; for that reason it is an intrinsic structure. It is usually regarded as belonging to the underlying affine space, since it is also invariant by translations.
One can generalise the definition of Zariski topology to projective spaces, and so to any algebraic variety as subsets of these. The general case of the Zariski topology is based on the affine scheme and spectrum of a ring constructions, as local models.
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