Show that $O(n)$, the set of orthogonal $n \times n$ matrices, is not connected

Question

I want to show that $O(n)$, the set of orthogonal $n \times n$ matrices is not connected.

---

I know that a connected space $X$ does not split into disjoint non-empty open subsets, so to prove $O(n)$ is not connected I need to find disjoint non-empty open subsets that partition $O(n)$. But I have not been able to get any

---

Any help would be much appreciated

Answer 1

Hint :

The function $f:O(n)\to \{-1,1\}$ defined by $f(A)=\operatorname{ det}(A)$ is continuous.