Can someone provide an example of a collection of matrix that is an open set? I am totally confused what it means for a matrix to form an open set. Open set to me is either an interval ( , ) in $\mathbb{R}$ or some dotted circle in $\mathbb{C}$ (ok, the dotted circle being $\{z \in \mathbb{C}: |z|<1\}$)
Can someone provide an intuitive explanation why and how certain collection of matrices form open set?
 A: $GL_n(\Bbb{R})$ the linear group of invertible matrices is an open set in $\mathcal{M}_n(\Bbb{R})$ the space of $n\times n$ matrices because it is the inverse image of $\Bbb{R}^*=\{x\in \Bbb{R},\,x\neq0\}$ which is open in $\Bbb{R}$, by a continuous (polynomial) function namely the determinant.
A: $GL_{n}(\mathbb{R})$ is an open subset of $M_{n}(\mathbb{R})$, where $M_{n}(\mathbb{R})$ is the set of all $n\times n$ matrices with real entries, and $GL_{n}(\mathbb{R})$ is called the general linear group which consists of invertible matrices.
Consider the determinant of a matrix as a map;that is,
$$det:GL_{n}(\mathbb{R})\rightarrow \mathbb{R}\setminus \left \{0  \right \}$$
The map is continuous and $\mathbb{R}\setminus \left \{0  \right \}$ is open in $\mathbb{R}$.Hence $GL_{n}(\mathbb{R})$ is open in $M_{n}(\mathbb{R})$.
Similarly, $SO_{n}(\mathbb{R})$ is an open subset of $O_{n}(\mathbb{R})$, where $O_{n}(\mathbb{R})$ and $SO_{n}(\mathbb{R})$ are called the orthogonal group and special orthogonal group respectively.
Again, consider the map
$$det:O_{n}(\mathbb{R})\rightarrow \left \{ \pm 1 \right \}$$
The inverse image of 1 by this map is $SO_{n}(\mathbb{R})$.Since the point 1 is open in $\left \{ \pm 1 \right \}$ we have that $SO_{n}(\mathbb{R})$ is an open subset of $O_{n}(\mathbb{R})$.
Again, $SL_{n}(\mathbb{R})$ is a closed subset of $M_{n}(\mathbb{R})$ since {1} is closed in $\mathbb{R}$.
