TITLE: Show that the set of all $n \times n$ orthogonal matrices, $O(n)$, is a compact subset of $\mbox{GL} (n,\mathbb R)$
QUESTION [11 upvotes]: I have only concept in topology, metric space, and functional analysis. How do I tackle this? Also  I want to know that is the set connected?

REPLY [2 votes]: In fact, you can say more: 
Using the canonical action of $SO(n)$ on $\mathbb{S}^{n-1}$, you can show that $SO(n)$ is homeomorphic to $\mathbb{S}^{n-1}$. But you have the exact sequence $1 \to SO(n) \to O(n) \to \mathbb{Z}_2 \to 1$, so $[O(n):SO(n)]=2$ and $O(n)$ is homeomorphic to $\mathbb{S}^{n-1} \coprod \mathbb{S}^{n-1}$.