0
$\begingroup$

Bourbaki's book on general topology states that a uniform space is metrizable if it is Hausdorff and the filter of entourages of the uniformity has a countable basis. However, he doesn't provide an example of a uniform space that is not metrizable. So I'm looking for examples of uniform spaces that aren't metrizable. If there are no important examples, then I don't see the motivation to study uniform spaces. I appreciate your help!

Improve this question
New contributor
RataMágica is a new contributor to this site. Take care in asking for clarification, commenting, and answering. Check out our Code of Conduct.
$\endgroup$
7
  • 6
    $\begingroup$ An infinite-dimensional Banach space (say, $\ell^2$) equipped with the weak topology has a natural uniform space structure, and it is not metrizable. $\endgroup$
    – David Gao
    Commented yesterday
  • $\begingroup$ @DavidGao, do you have some bibliography where I can study that with detail? $\endgroup$
    – RataMágica
    Commented 23 hours ago
  • 1
    $\begingroup$ It is known that every compact Hausdorff topological space admits exactly one uniformity (induced by all (finite) open covers). The notion of metrizability of a compact Hausdorff topological space agrees with metrizability of the corresponding uniform space. Hence any nonmetrizable compact Hausdorff topological space answers your question. Such spaces can be build up e.g. using Tychonoff theorem (as uncountable products) or as Čech-Stone compactification of a noncompact Tychonoff space. $\endgroup$
    – Benjamin Vejnar
    Commented 10 hours ago
  • 1
    $\begingroup$ Every topological group comes with several useful uniform structures (left, right, etc). They are metrizable if and only if the topology of the group is. If you want to study details, you can start with "Uniform structures on topological groups and their quotients" by Roelcke and Dierolf (1981). $\endgroup$
    – user95282
    Commented 3 hours ago
  • 2
    $\begingroup$ For more examples and motivation, see "Uniform ideas in analysis" by M.D. Rice, Real Analysis Exchange 6 (1980/81), 139--185. $\endgroup$
    – user95282
    Commented 3 hours ago

0

Reset to default

You must log in to answer this question.

Browse other questions tagged .