Is there a relation between coarseness and metrizability?

[quasar987]

For instance, if a topology A is metrizable and either

(i) B is coarser than A

or

(ii) B is finer than A.

Can we say something the metrizability of B?

 

[morphism]

Take the discrete and indiscrete topologies on an infinite set. The former is finer than the latter, but only the former is metrizable.

Conversely, take R with its usual topology and the Sorgenfrey (lower limit) topology. The former is coarser than the latter, but, again, only the former is metrizable.

So the answer is no - there is no real relation.