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Products and the Hausdorff Axiom

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Part of the Lecture Notes in Mathematics book series (LNM,volume 1358)

Abstract

We want to define the product X×Y of any two prevarieties X,Y. Now we will certainly want to have An × Am ≅ An+m. But the product of the Zariski topologies in An and Am does not give the Zariski topology in An+m; in A1 × A1, for instance, the only closed sets in the product topology are finite unions of horizontal and vertical lines. The only reliable way to find the correct definition is to use the general category-theoretic definition of product.

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© 1988 Springer-Verlag Berlin Heidelberg

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Mumford, D. (1988). Products and the Hausdorff Axiom. In: The Red Book of Varieties and Schemes. Lecture Notes in Mathematics, vol 1358. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-21581-4_6

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  • DOI: https://doi.org/10.1007/978-3-662-21581-4_6

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-50497-9

  • Online ISBN: 978-3-662-21581-4

  • eBook Packages: Springer Book Archive