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On the number of locally bounded field topologies

Author: Jo-Ann D. Cohen
Journal: Proc. Amer. Math. Soc. 90 (1984), 207-210
MSC: Primary 12J05
MathSciNet review: 727234
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Abstract: Kiltinen has proven that there exist $ {2^{\left\vert F \right\vert}}$ first countable, locally bounded field topologies (the maximum number possible) on a field $ F$ of infinite transcendence degree over its prime subfield. We consider those fields $ F$ of countable transcendence degree over its prime subfield $ E$. In particular it is shown that if the characteristic of $ F$ is zero and the transcendence degree of $ F$ over $ E$ is nonzero or if $ F$ is a field of prime characteristic and the transcendence degree of $ F$ over $ E$ is greater than one, then there exist $ {2^{\left\vert F \right\vert}}$ normable, locally bounded field topologies on $ F$.

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