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The Basic Zariski Topology
Davide Rinaldi1 ; Giovanni Sambin2 ; Peter Schuster3
1 Department of Pure Mathematics, University of Leeds, Leeds LS2 9JT, England
2 Dipartimento di Matematica, Università di Padova, Via Trieste 63, 35121 Padova, Italy
3 Dipartimento di Informatica, Università di Verona, Strada le Grazie 15, 37134 Verona, Italy
Confluentes Mathematici, Tome 7 (2015) no. 1, pp. 55-81.

We present the Zariski spectrum as an inductively generated basic topology à la Martin-Löf and Sambin. Since we can thus get by without considering powers and radicals, this simplifies the presentation as a formal topology initiated by Sigstam. Our treatment includes closed subspaces and basic opens: that is, arbitrary quotients and singleton localisations. All the effective objects under consideration are introduced by means of inductive definitions. The notions of spatiality and reducibility are characterized for the class of Zariski formal topologies, and their nonconstructive content is pointed out: while spatiality implies classical logic, reducibility corresponds to a fragment of the Axiom of Choice in the form of Russell’s Multiplicative Axiom.

DOI : 10.5802/cml.18
Classification : 03E25, 03F65, 13A99, 54B35
Davide Rinaldi 1 ; Giovanni Sambin 2 ; Peter Schuster 3

1 Department of Pure Mathematics, University of Leeds, Leeds LS2 9JT, England
2 Dipartimento di Matematica, Università di Padova, Via Trieste 63, 35121 Padova, Italy
3 Dipartimento di Informatica, Università di Verona, Strada le Grazie 15, 37134 Verona, Italy 
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Davide Rinaldi; Giovanni Sambin; Peter Schuster. The Basic Zariski Topology. Confluentes Mathematici, Tome 7 (2015) no. 1, pp. 55-81. doi : 10.5802/cml.18. https://cml.centre-mersenne.org/articles/10.5802/cml.18/

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