THE SPACE OF R-PLACES OF R(x, y) IS NOT METRIZABLE
نویسنده
چکیده
For n = 1, the space of R-places of the rational function field R(x1, . . . , xn) is homeomorphic to the real projective line. For n ≥ 2, the structure is much more complicated. We prove that the space of R-places of the rational function field R(x, y) is not metrizable. We explain how the proof generalizes to show that the space of R-places of any finitely generated formally real field extension of R of transcendence degree ≥ 2 is not metrizable. We also consider the more general question of when the space of R-places of a finitely generated formally real field extension of a real closed field is metrizable.
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