On the number of field topologies on an infinite field

Kiltinen, John O.

doi:10.1090/S0002-9939-1973-0318118-4

On the number of field topologies on an infinite field
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Proc. Amer. Math. Soc. 40 (1973), 30-36 Request permission

Abstract:

K.-P. Podewski has recently proven that every countable infinite field admits ${2^{{}_2{\aleph _0}}}$ different field topologies. Using methods of valuation theory, it is proven that every uncountable field, and more generally, every field $F$ of infinite transcendence degree over some subfield, admits ${2^{{2^{|F|}}}}$ field topologies. By purely set theoretic considerations, it then follows that there are ${2^{{2^{|F|}}}}$ field topologies on any infinite field $F$, no two of which are topologically isomorphic. This latter result is then generalized to any infinite commutative ring without proper zero-divisors. A further aspect of Podewski’s work on countable fields is generalized in a final theorem which states that a field $F$ of infinite transcendence degree admits ${2^{{2^{|F|}}}}$ field topologies which fail to be suprema of locally bounded ring topologies.

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Über zwei Sätze von G. Fichtenholz und L. Kantorovich

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Cardinal and ordinal numbers

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