Let $V_k(n) \subset \prod _{i=1} ^k S^n$ the real Stiefel space endowed with subspace topology and defined via
$$V_k(n) := \{(v_1, v_2, ..., v_k) \vert \text{ } v_i \bot v_j \text{ for } i \neq j \text{ and } \left\| v_i \right\|=1 \}$$
I know that there are a lot of ways to show that $V_k(n)$ is a manifold but I intend to prove it using the group action by $O(k)$ and to show that $V_k(n) \cong O(n)/O(n-k)$.
The problem is to show that $O(k)/O(n-k)$ is a manifold. The concrete point I'm struggle now is to verify the totally discontinuity of the action.
Therefore I don't know how to show that for every $(v_1, v_2, ..., v_k) \in V_k(n)$ there exist an open neightbourhood $U$ such that for all $f \neq g \in O(n)$ we have $fU \cap gU = \emptyset$.
The problem is that $O(k)$ is not discrete so I suppose that I need a clever choice of $U$, but I don't find the correct choice.