Lately I've been delving into the Lorentz and Poincaré groups and their Lie algebras, as well as Lie groups in general. I have a good understanding of the (infinitesimal) generators of boosts and translations, but I am still at a loss regarding the topologies (open sets) of Lie groups of matrix spaces and of general pseudo-Euclidean spaces.
Given a vector norm $\lVert \cdot\rVert:V\to\mathbb R$, does the operator norm of a matrix
$$\lVert A\rVert_{op}=\sup{\left\lbrace\lVert A x\rVert : \lVert x\rVert \leq 1\right\rbrace}$$
induce a useful or "standard" topology on the space $\mathbb{R}_{m \times n}$ of $m\times n$ matrices? Also, given a vector space isomorphism $f:\mathbb{R}^{m\cdot n}\to \mathbb{R}_{m\times n}$, does the final topology
$$\mathcal{T}=\left\lbrace U \subseteq \mathbb{R}_{m\times n} : f^{-1}(U) \text{ open}\right\rbrace$$
define another useful topology? Furthermore, in a pseudo-Euclidean space $\mathbb{R}^{p,q}$, does a quadratic form $q:\mathbb{R}^{p,q}\to \mathbb{R}$ (or its induced symmetric bilinear form) induce any kind of topology?
In a broader sense, how does one go about showing that $\mathrm{O}(p,q)$, $\mathrm{SO}(p,q)$, $\mathrm{GL}(n)$, etc. are Lie groups i.e. differentiable manifolds?