Prove the general linear group is open in $\mathbb{R}^{n^2}$ I was wondering if somebody could help me with the following problem. 
Let $GL(n,\mathbb{R})$ be the space of $n \times n$ matrices over $\mathbb{R}$. Prove that $GL(n,\mathbb{R})$ identified in the standard way with a subset $E(n^2)\cong \mathbb{R}^{n^2}$, is open in $\mathbb{R}^{n^2}$. 
I was thinking of defining a map , sending every real $n \times n$ matrix to its determinant, but I'm entirely sure this works. Any help would be greatly appreciated!
Additional Question: Prove that $GL(n,\mathbb{R})$ with the relative topology from $\mathbb{R}^{n^2}$, is not connected

To show that $GL(n,\mathbb{R})$ is open in the space of all $n\times n$ matrices $E(n^2) \cong \mathbb{R}^{n^2}$, your idea is good:
The determinant
$$ \text{det}: E(n^2) \to \mathbb{R} $$
is continuous because for any matrix $A\in E(n^2)$, $\text{det}(A)$ can be written as a polynomial in the entries of $A$. We have
$$ GL(n,\mathbb{R}) = \text{det}^{-1}(\mathbb{R}\setminus\{0\}), $$
and since $\mathbb{R}\setminus\{0\}$ is open in $\mathbb{R}$, it follows that $GL(n,\mathbb{R})$ is open in $E(n^2)$.
