Let M be the k*n matrices of rank k(k <= n),O= { A is in M: A^{t}A=I }.
How to prove O is compact in M?
The set of $\displaystyle k\times n$ matrices can by identified in the obvious way with $\displaystyle \mathbb{R}^{nk}$, and can be given the standard metric topology. Heine-Borel applies in this setting, too. Bounded is easy here, and for closed, here's a hint: is the map $\displaystyle A\mapsto AA^T$ continuous?