well, In Brian C Halls Book, I am not getting the definition of Matrix Lie group, as he says : A matrix Lie Group is any subgroup $G$ of $GL_n(\mathbb{C})$ with the following property: If $A_m$ is any sequence of matrix in $G$ and and $A_m$ converges to $A$ then either $A\in G$ or $A$ is not invertible. well, That may happen true, but after that page he gave an example of a subgroupof $GL_n(\mathbb{C})$ which is not closed hence not a matrix lie group! could any one tell me clearly what is the definition?
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$\begingroup$The essence of that definition is that the subgroup be closed as a subset of the topological space $GL_n(\Bbb C)$. It will usually not be closed as a subset of the space of all matrices (although it might in some cases be) because $GL_n(\Bbb C)$ itself is not. This is why the exception made is for sequences converging to a non-invertible matrix: within the space $GL_n(\Bbb C)$ such sequences are not convergent at all, so need not be considered for the question of closure.
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