Is there a relation between coarseness and metrizability?

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Metrizability does not have a direct correlation with coarseness or fineness of topologies. A metrizable topology can have a coarser or finer topology that is not metrizable, as illustrated by the discrete and indiscrete topologies on an infinite set. Similarly, the usual topology on R is coarser than the Sorgenfrey topology, yet only the usual topology is metrizable. Therefore, the relationship between coarseness, fineness, and metrizability is not established. In conclusion, the discussion concludes that there is no real relation between these properties.
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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?
 
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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.
 

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