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J. Math. 87 (1965), 605–630, 631–648. [Ax and Kochen 1966] J. Ax and S. Kochen, “Diophantine problems over local fields, III”, Ann. of Math. (2) 83 (1966), 437–456. [Denef and van den Dries 1988] J. Denef and L. van den Dries, “p-adic and real subanalytic sets”, Ann. of Math. (2) 128:1 (1988), 79–138. [van den Dries 1998] L. van den Dries, Tame topology and o-minimal structures, London Math. Soc. Lecture Note Series 248, Cambridge Univ. Press, Cambridge, 1998. [van den Dries and Miller 1996] L.

Lecture Note Series 248, Cambridge Univ. Press, Cambridge, 1998. [van den Dries and Miller 1996] L. van den Dries and C. Miller, “Geometric categories and o-minimal structures”, Duke Math. J. 84:2 (1996), 497–540. [van den Dries and Speissegger 1998a] L. van den Dries and P. Speissegger, “The real field with convergent generalized power series”, Trans. Amer. Math. Soc. 350:11 (1998), 4377–4421. [van den Dries and Speissegger 1998b] L. van den Dries and P. Speissegger, “The field of reals with multisummable series and the exponential function”, preprint, 1998.

Is compact. This shows that the Zariski topology on Spec(F [X]) ¯ but the Stone topology We can identify (as objects) Sn (F ) and Spec(F [X]), is much finer that the Zariski topology. The Stone topology corresponds to the topology generated by the constructible sets. If p ∈ Sn (F ), let V = {x ∈ K n : f (x) = 0 for all f ∈ Ip }. Then the type p asserts that x ¯ ∈ V and x ¯ ∈ W for any W ⊂ V defined over F . Thus realizations of p are points of V generic over F . What about real closed fields?

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