Defining Lie groups without the notion of a manifold I like to introduce (matrix) Lie groups without the notion of manifolds. (And I am happy to sacrify the "few" Lie groups which are not matrix groups in favor of a simpler definition.) I was thinking of the following definition: * $G$ is a (matrix) Lie groups $:\Leftrightarrow$ $G$ is a closed subgroups of $GL(n,\mathbb{R})$. (I do not care about Lie groups over finit fields either...) This definition seems to be okay for my purpose but it requires to equip $GL(n,\mathbb{R})$ with a metric (to give closeness a meaning). My question: Am I correct: If I do not want to use the notion of a manifold (or a non-standard replacement with similar complexity), I need to equip $GL(n,\mathbb{R}$ with a metric to sufficiently characterize matrix Lie groups?

You can take the approach given in Hall's book on Lie groups where he defines a matrix Lie group to be a closed subgroup of $\text{GL}_n(\Bbb{C})$. Of course when we mean closed it is with respect to the usual Euclidean topology on $\Bbb{C}^{n^2}$.

You may choose not to equip $\text{GL}_n(\Bbb{C})$ with a metric but I think ultimately at the end of the day you want to equip it with at least some kind of _topology_. From my experience the usual topology on these groups is the one induced from the Euclidean metric.