Is there a relation between coarseness and metrizability?

1. Mar 2, 2008

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?

2. Mar 2, 2008

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.