On the number of topologies on countable fields

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.