Consider the real vector space space $\mathrm{M}(2,\Bbb{R})$ of $2\times 2$ real matrices. Identifying $\mathrm{M}(2,\Bbb{R})$ with $\Bbb{R}^4$ with the standard Euclidean metric and the corresponding metric topology, we observe that $\mathrm{GL}(2,\Bbb{R})$ is an open subset of $\mathrm{M}(2,\Bbb{R})$. Let $Y$ be an open subset of $\mathrm{GL}(2,\Bbb{R})$, such that $A,B\in Y$ implies that $AB^{-1}\in Y$. Then prove that $Y$ is closed in $\mathrm{GL}(2,\Bbb{R})$.
Can someone give me some hints about this problem, how to solve it?
Any help will be highly appreciated.
Recall some elementary algebra: $Y$ is a subgroup of $\text{GL}(2,\Bbb R)$. The latter forms a topological group, as the group operations (multiplication and inversion) are continuous.
An open subgroup of a topological group is closed, because its complement is a union of cosets which are also open (being translates of the subgroup).