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## On the number of topologies on countable fields

### Abstract

For any countable field $R$ and any non-discrete metrizable field topology $\tau _0$ of the field, the lattice of all field topologies of the field admits: \newline -- Continuum of non-discrete metrizable field topologies of the field stronger than the topology $\tau _0$ and such that $\sup \{\tau _1, \tau _2 \}$ is the discrete topology for any different topologies; \newline -- Continuum of non-discrete metrizable field topologies of the field stronger than $\tau _0$ and such that any two of these topologies are comparable; \newline -- Two to the power of continuum of field topologies of the field stronger than $\tau _0$, each of them is a coatom in the lattice of all topologies of the field.

V.I. Arnautov
Institute of Mathematics and Computer Science