Is there a relation between coarseness and metrizability?

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The discussion concludes that there is no definitive relationship between the coarseness or fineness of topologies and their metrizability. Specifically, it highlights that while a finer topology, such as the discrete topology, is metrizable, a coarser topology, like the indiscrete topology, is not. Additionally, it points out that in the case of the real numbers with the usual topology and the Sorgenfrey topology, the former is coarser yet metrizable, reinforcing the lack of correlation 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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