What Are the Open Sets of U(N)?

[smallgun]

Hi people,

Let $U(N)$ be the unitary matrices group of a positive integer $N$.

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

I am curious what the open sets of $U(N)$ are in this case. If it has an inherited topology from $GL(N,\mathbb{C})$, what are the open sets of $GL(N,\mathbb{C})$? 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, $U(N)$ can be viewed as a subspace of $\mathbb{R}^{2N^2}$.

I am curious what the open sets of $U(N)$ 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.