closed subgroup

《Class Field Theory》 · 印刷第 4 页 Section 1.1，Definition 1.2 的矩阵空间记号、Definition 1.4 及紧接其后的闭子群等价说明

Let $M_n(\mathbb C)$ denote the space of all $n\times n$ matrices with complex entries. […] Definition 1.4. A matrix Lie group is a subgroup $G$ of $\operatorname{GL}(n;\mathbb C)$ with the following property: If $A_m$ is any sequence of matrices in $G$, and $A_m$ converges to some matrix $A$, then either $A$ is in $G$ or $A$ is not invertible. The condition on $G$ amounts to saying that $G$ is a closed subset of $\operatorname{GL}(n;\mathbb C)$. (This does not necessarily mean that $G$ is closed in $M_n(\mathbb C)$.) Thus, Definition 1.4 is equivalent to saying that a matrix Lie group is a closed subgroup of $\operatorname{GL}(n;\mathbb C)$.