What are the open sets of U(N)?

[smallgun]

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.

 

[Eynstone]

U(N) is metrizable as it inherits the metric from R^N^2.

 

[Landau]

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.

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.