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.