Is $\Omega=D_n(K) \cap Gl_n(K) $an open set?

Question

let $D_n(K)$ the set of diagonalizable matrix and $Gl_n(K)$ the set of inversible ones.

is $\Omega=D_n(K) \cap Gl_n(K) $ an open set ?

I can see that $D_n(K)$ is not an open set and $Gl_n(K)$ is open but that doesnt show anything.

Answer 1

No, the set of diagonalizable invertible matrices is not open. Consider
the matrix $$ \begin{pmatrix} 1 &\epsilon\\ 0 & 1 \end{pmatrix} , $$ which is diagonalizable if and only if $\epsilon=0.$