Let $U$ be the set of all $n×n$ matrices with real entries such that all their eigenvalues belong to $\mathbb C \setminus \mathbb R $, and $X = M_n(\mathbb R)$. Is $U$ open?
For $n$ odd $U$ is open. Since characteristic polynomial is of odd degree, Hence it must be having at least one real zero. So, $U=\emptyset$. Hence, $U$ is open. What about the other case? When $n$ is even, How do I connect characteristic polynomial and nature of roots? For quadratic equations, it is an easy task.
Let $A \in U$. If $A$ is not an interior point of $U$ then there exists matrices $A_n$ converging to $A$ such that each $A_n$ has a real eigen value $\lambda_n$. There exist unit vectors $x_n$ such that $A_nx_n=\lambda_n x_n$. Since the unit sphere is compact there is a subsequence $(x_{n'})$ converging to a unit vector $x$. Now, $A_{n'}x_{n'}\to Ax$ and the equation$A_{n'}x_{n'}=\lambda_{n'}x_{n'}$ shows that $\lambda_{n'}$ must converge. The limit $\lambda$ is a real number and we get the contradiction $Ax=\lambda x$.