Is the group $\mathcal{U} (n)$ of all $n\times n$ unitary matrices over $\mathbb{C}$ a (local) path-connected space? 


If so, what are the connected components of the unitary matrix group $\mathcal{U}(n)$? Is the number of components finite? What is the representative for each component? Is each components closed in norm topology?



Is the unitary matrix group path-connected?
Every matrix Lie group is a smooth manifold, hence it is path-connected if and only if it is connected.
And $U(n)$ is compact and connected as a topological space (any unitary matrix can be diagonalized by a unitary matrix, this gives a path from it to the identity). It is not simply-connected, though.
We have $\pi\_1(U(n))\simeq \mathbb{Z}$.


