This is a question of mathematical nature. Let the Lorentz group $O(1,3)$ be defined as a matrix group.
$$\text{O}(1,3) =\{\Lambda\in M_4 (\mathbb{R})| \Lambda^T \text{diag}(+---) \Lambda = \text{diag}(+---)\} $$
One defines the supremum norm on $M_4 (\mathbb{R})$ as:
which turns $M_4 (\mathbb{R})$ into a topological space in the metric topology induced by the norm. Can this supremum norm be made particular for Lorentz matrices? I guess it would, right? It would immediately follow that the Lorentz group is a topological space in the subspace topology of $M_4 (\mathbb{R})$.