Let $X$ be a topological space and let $U\subset X$ . In which of the following cases is $U$ open ?
a)Let $U $ be the set of invertible upper triangular matrices in $\mathbb{M_n(\mathbb{R})}$, where $n\ge 2$ and $X=\mathbb{M_n(\mathbb{R})}$
b)Let $U$ be the set of all $2\times 2$ matrices with real entries such that al their eigenvalues belong to $\mathbb{C}$\ $\mathbb{R}$ , and $X=\mathbb{M}_n(\mathbb{R})$
c)Let $U$ be the set of all complex numbers $\lambda$ such that $A-\lambda I$ is invertible , where $A$ is a given $3\times 3$ matrix with complex entries , and $X=\mathbb{C}$
for option (b) i think it would be open as for complex eigenvalues we have the discreminant of characteristic polynomial is strictly less than $0$ so open . but no idea about (a) and (c)
Hint
a) Just for simplicity assume $n=2.$ An element of $U$ is as $$\begin{pmatrix}a& b\\ 0 & c\end{pmatrix}$$ with $ac\ne 0.$ Identify $M_ 2(\mathbb{R})$ with $\mathbb{R}^4.$ Can you find a neigbhourhood of $(a,b,0,c)$ wich is a subset of $U?$
c) $\lambda \in \mathbb{C}\to \det(A-\lambda I)\in\mathbb{C}$ is continuous. What can you say about $\det^{-1}(0)?$