Let $A \in \mathscr{M}_n(\mathbb{C})$. We denote by $\mathbb{C}[A]$ the set of elements of $\mathscr{M}_n(\mathbb{C})$ of the form $P(A)$ where $P \in \mathbb{C}[X]$ is a polynomial. We denote $$(\mathbb{C}[A])^* = \left\{B \in \mathbb{C}[A] \cap \mathrm{GL}_n(\mathbb{C}) \mid B^{-1} \in \mathbb{C}[A]\right\}$$ We want to show that $\exp(\mathbb{C}[A]) = (\mathbb{C}[A])^*$. We suppose that $\exp(\mathbb{C}[A]) \neq (\mathbb{C}[A])^*$ and we fix $M_1, M_2 \in (\mathbb{C}[A])^*$ such that $M_1 \in \exp(\mathbb{C}[A])$ and $M_2 \notin \exp(\mathbb{C}[A])$. Show that there exists a continuous map $f$ from $(\mathbb{C}[A])^*$ to $\{0,1\}$ such that $f(M_1) = 0$ and $f(M_2) = 1$.
Let $A \in \mathscr{M}_n(\mathbb{C})$. We denote by $\mathbb{C}[A]$ the set of elements of $\mathscr{M}_n(\mathbb{C})$ of the form $P(A)$ where $P \in \mathbb{C}[X]$ is a polynomial. We denote
$$(\mathbb{C}[A])^* = \left\{B \in \mathbb{C}[A] \cap \mathrm{GL}_n(\mathbb{C}) \mid B^{-1} \in \mathbb{C}[A]\right\}$$
We want to show that $\exp(\mathbb{C}[A]) = (\mathbb{C}[A])^*$. We suppose that $\exp(\mathbb{C}[A]) \neq (\mathbb{C}[A])^*$ and we fix $M_1, M_2 \in (\mathbb{C}[A])^*$ such that $M_1 \in \exp(\mathbb{C}[A])$ and $M_2 \notin \exp(\mathbb{C}[A])$.
Show that there exists a continuous map $f$ from $(\mathbb{C}[A])^*$ to $\{0,1\}$ such that $f(M_1) = 0$ and $f(M_2) = 1$.