Summary of the topology of $M_n(\mathbb{R})$

Identification. $M_n(\mathbb{R})\cong\mathbb{R}^{n^2}$ via entries; all norms are equivalent.

Convergence. Entrywise convergence $=$ convergence for any matrix norm.

Completeness. $M_n(\mathbb{R})$ is a Banach space.

Compactness. Heine–Borel: compact $\iff$ closed and bounded. $O_n$ is compact (closed + bounded); $GL_n$, $SL_n$, $S_n$ are closed or open subsets as appropriate.

Subgroups. $GL_n$ is open; $SL_n$, $O_n$, $SO_n$ are closed; $O_n$, $SO_n$ are compact.

Matrix exponential. $\exp$ is continuous, even locally Lipschitz.

Eigenvalues. Depend continuously on the matrix entries.

Separability. $M_n(\mathbb{Q})$ is a countable dense subset.

Conclusion. $M_n(\mathbb{R})$ inherits from $\mathbb{R}^{n^2}$ all its metric properties; the classical matrix groups add rich geometric structure.