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Math Help - Weakest Metrizable Topology?

  1. #1
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    Weakest Metrizable Topology?

    Given a set X and a nonmetrizable topology \mathcal{T}, is the intersection \bigcap\left\{\mathcal{U}\subseteq\mathcal{P}(X)| \mathcal{U}\text{ is metrizable and }\mathcal{T}\subset\mathcal{U}\right\} metrizable?

    If not, are there properties that \mathcal{T} could have that would make that intersection metrizable?
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Re: Weakest Metrizable Topology?

    Quote Originally Posted by SlipEternal View Post
    Given a set X and a nonmetrizable topology \mathcal{T}, is the intersection \bigcap\left\{\mathcal{U}\subseteq\mathcal{P}(X)| \mathcal{U}\text{ is metrizable and }\mathcal{T}\subset\mathcal{U}\right\} metrizable?

    If not, are there properties that \mathcal{T} could have that would make that intersection metrizable?
    What have you tried? It seems like you could, if its true, prove this using Nagata-Smirnov.
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    Re: Weakest Metrizable Topology?

    That looks like an excellent place to start. Thank you. I will get back to you with any results that I find.
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