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Is there a relation between coarseness and metrizability?

  1. Mar 2, 2008 #1

    quasar987

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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?
     
  2. jcsd
  3. Mar 2, 2008 #2

    morphism

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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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