Abstract
Suppose that our algebraically closed ground field k is given a topology making it into a topological field. The most interesting case of this is when k = ℂ, the complex numbers. However, we can make at least the first definition in complete generality. Namely, I claim that when k is a topological field, then there is a unique way to endow all varieties X over k with a new topology, which we will call the strong topology, such that the following properties hold:
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i)
the strong topology is stronger than the Zariski-topology, i.e., a closed (resp. open) subset Z ⊂ X is always strongly closed, (resp. strongly open).
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ii)
all morphisms are strongly continuous
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iii)
if Z ⊂ X is a locally closed subvariety, then the strong topology on Z is the one induced by the strong topology on X
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iv)
the strong topology on X × Y is the product of the strong topologies on X and on Y
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v)
the strong topology on Δ l is exactly the given topology on k.
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© 1988 Springer-Verlag Berlin Heidelberg
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Mumford, D. (1988). Complex varieties. 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_10
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DOI: https://doi.org/10.1007/978-3-662-21581-4_10
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-50497-9
Online ISBN: 978-3-662-21581-4
eBook Packages: Springer Book Archive
