$X \subseteq M(n,\mathbb C) ; |X|>1 ; $ connected/path connected, what about $S:=\{x \in \mathbb C : x $ is an eigenvalue of some matrix in $X\}$?

Question

Let $X \subseteq M(n,\mathbb C)$ be a set with more than one element and $S:=\{x \in \mathbb C : x $ is an eigenvalue of some matrix in $X\}$. I know that if $X$ is compact then so is $S$. My question is:

If $X$ is connected, then is $S$ also connected ? What if $X$ is path connected ?

Please help, thanks in advance.

Answer 1

Take $$X = \{{\rm{diag}}(x, -x) \,|\, x \in (1,2)\}.$$ Then $X$ is path-connected, but $S = (1,2) \cup (-2,-1)$ is not.