On the number of topologies on countable fields

V. I. Arnautov ; G. N. Ermakova

Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 1 (2019), pp. 79-90.

Résumé

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: – 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; – Continuum of non-discrete metrizable field topologies of the field stronger than ${\tau }_0$ and such that any two of these topologies are comparable; – 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.

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