I am reading through this paper (the Eastin-Knill Theorem) and there is a step in the proof of the main theorem that I do not understand.
Let $Q$ be a composite quantum system supporting a quantum code $C$. For a quantum system of finite dimension $d$, let $U(d)$ denote the set of unitaries on that quantum system. For any $d$, $U(d)$ is a compact connected Lie group. Let $\mathcal{T}$ be the set of unitary product operators on $Q$. Being a direct product of compact and connected Lie groups, $\mathcal{T}$ is also compact and connected. Let $\mathcal{G}$ be the set of unitary product operators that are also logical operators with respect to the code $C$ on $Q$. It is shown in the paper that $\mathcal{G}$ is a Lie subgroup of $\mathcal{T}$. As a Lie group, $\mathcal{G}$ can be partitioned into cosets of the connected component of the identity, $\mathcal{C}$. The set of cosets is $R = \mathcal{G}/ \mathcal{C}$ which constitutes a topologically discrete group.
Let $\mathcal{F}$ be a set consisting of one representative from each coset of $\mathcal{C}$ in $\mathcal{G}$, that is, one element from each set in $R$. Since $R$ is discrete, it is obvious that $\mathcal{F}$ is discrete as well. The authors then claim that $\mathcal{F}$ is finite since it is a discrete subset of a compact set, namely $\mathcal{T}$. But this alone is not enough to guarantee finiteness (consider the discrete subset $\{\frac{1}{n}\}_n$ of $[0,1]$). If it were also (topologically) closed, that would be sufficient. But I cannot figure out a reason why this set must be closed, or any other reason why $\mathcal{F}$ might be finite.
Would greatly appreciate an explanation if anyone knows why.