Is the following subset of space of matrices connected, open? If $S=\left\{ A=\begin{bmatrix}A_1&0 \\ 0&A_2  \end{bmatrix} \in \mathbb{M}_4(\mathbb{C}): det A_1=detA_2 \right\}$. Then is this set open in $\mathbb{M}_{4}(\mathbb{C})$ with usual topology. If not then what are the interior points. I would like some hints towards this. Also if someone could suggest a book to study topology of space of complex matrices, it will be helpfull to me. Thank you. 
I have an analytic map from open unit disc to the norm unit ball of space of matrices having image contained in $S$. That is why i am wondering about the interior of this set. 

An idea: denote by $\;\mathcal B\;$ the set of all blocks-matrices in $\;M_4(\Bbb C)\;$, and define 
$$f:\mathcal B\to\Bbb C\;,\;\;f\begin{pmatrix}A_1&0 \\ 0&A_2  \end{pmatrix}:=\det A_1-\det A_2$$
The above map is continuous (wrt the usual topologies in domain and codomain: the Euclidean one in $\;\Bbb C\;$ and the one in $\;\mathcal B\subset M_4(\Bbb C)\;$ inherited from the Euclidean one in $\;\Bbb C^4\;$), as it is apolynomial map in the entries of the matrix $\;A=\begin{pmatrix}A_1&0 \\ 0&A_2  \end{pmatrix}\;$. 
Finally, just observe that $\;S=f^{-1}(\{0\})\;$ and thus $\;S\;$ is closed...
