On non-discrete topologization of some countable skew fields

V. I. Arnautov ; G. N. Ermakova

Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 1 (2021), pp. 84-92.

Résumé

If for any finite subset $M$ of a countable skew field $R$ there exists an infinite subset $S\subseteq R$ such that $r\cdot m=m\cdot r$ for any $r\in S$ and for any $m\in M$, then the skew field $R$ admits: – A non-discrete Hausdorff skew field topology ${\tau }_0$. – Continuum of non-discrete Hausdorff skew field topologies which are stronger than the topology ${\tau }_0$ and such that $sup\{{\tau }_1,{\tau }_2\}$ is the discrete topology for any different topologies ${\tau }_1$ and ${\tau }_2$; – Continuum of non-discrete Hausdorff skew field topologies which are stronger than ${\tau }_0$ and such that any two of these topologies are comparable; – Two to the power of continuum Hausdorff skew field topologies stronger than ${\tau }_0$, and each of them is a coatom in the lattice of all skew field topologies of the skew fields.

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