Demonstrate that the set of orthonormal matrices is closed

I'm trying to demonstrate that the set of orthonormal matrices $$\mathcal{O}(n,m) =\{ A | A^TA = I_m\}$$ is closed. I am in a metric space $$\mathbb{R}^{n*p}$$ that uses the measure $$d^2(X,Y) = trace((X-Y)^T(X-Y))$$. I don't really understand how to interpret the notion of closed when it comes to this example since it's with matrices and not singular values, functions, or open/closed balls.

Let's denote $$\mathcal M(n,m)$$ the linear space of real matrices of dimension $$n \times m$$. The map
$$\begin{array}{l|rcl} \phi : & \mathcal M(n,m) & \longrightarrow & \mathcal M(m,m) \\ & A & \longmapsto & A^tA \end{array}$$
is continuous as $$A^tA$$ entries are polynomial functions of $$A$$ entries.
The singleton $$\{I_m\}$$ is a closed subset of the normed vector space $$(\mathcal M(m,m), \Vert \cdot \Vert_2)$$ where $$\Vert \cdot \Vert_2$$ stands for the norm defined by
$$\Vert A \Vert_2^2 = \sum_{i,j} a_{ij}^2.$$ Therefore, $$\mathcal O(n,m)$$ which is the inverse image of $$\{I_m\}$$ under $$\phi$$ is closed for the topology induced by the distance $$d$$ as
$$d^2(X,Y) = \Vert X-Y \Vert_2^2.$$