Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

What are the open sets of U(N)?

  1. Aug 5, 2011 #1
    Hi people,

    Let [itex] U(N) [/itex] be the unitary matrices group of a positive integer [itex] N [/itex].

    Then, [itex] U(N) [/itex] can be viewed as a subspace of [itex] \mathbb{R}^{2N^2} [/itex].

    I am curious what the open sets of [itex] U(N) [/itex] are in this case. If it has an inherited topology from [itex] GL(N,\mathbb{C}) [/itex], what are the open sets of [itex] GL(N,\mathbb{C}) [/itex]? I know by the definition of a topological group the two maps, matrix multiplication and inverse, should be continuous. Can we deduce the open sets from those two maps?

    Thank you for reading my question.
     
    Last edited: Aug 5, 2011
  2. jcsd
  3. Aug 10, 2011 #2
    U(N) is metrizable as it inherits the metric from R^N^2.
     
  4. Aug 20, 2011 #3

    Landau

    User Avatar
    Science Advisor

    You just said it yourself. View U(n) as subspace of R^{2n^2}. You know the open sets of R^{2n^2}, hence of every subspace of it.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: What are the open sets of U(N)?
Loading...