Let $\pi: \mathbb R^n \to \mathcal P_n$ denote the bijection between the coefficients and the real monic $n^{\text{th}}$ degree polynomials. That is, for $a = (a_0, \dots, a_{n-1}) \in \mathbb R^n$, $\pi(a) = x^n + a_{n-1} x^{n-1} + \dots + a_0$.
Let us denote $$ \Delta = \{ v \in \mathbb R^n: \pi(v) \text{ has all roots in the open unit disk of } \mathbb C\},$$ and $$ \Gamma = \{ v \in \mathbb R^n: \pi(v) \text{ has all roots in the closed unit disk of } \mathbb C\}.$$ I think we can show $\Delta$ is open and $\Gamma$ is closed in Eclidean topology. With Vieta's formula, we can also claim $\Delta$ and $\Gamma$ are both path connected. If I am not mistaken, we can also show $\overline{\Delta} = \Gamma$ where $\bar{\cdot}$ denotes closure of a set. Because for $w \in \Gamma \setminus \Delta$ with roots $r_1, \dots, r_n \in \mathbb C$, then \begin{align*} u(\alpha) := \pi^{-1}\left( (x-\alpha r_1) \dots (x-\alpha r_n) \right) \end{align*} is in $\Delta$ when $\alpha < 1$ and $\lim_{\alpha \to 1} u(\alpha) = w$.
My question is: can we claim $\Delta$ is regular open, that is the interior of $\Gamma$ is $\Delta$, i.e., $\Gamma^{\circ} = \Delta$?