Recall that a domain $D \subseteq \mathbb C$ is called *regular* if for each point $x \in \partial D$, we have $\mathbf P_x\lbrack \tau_D = 0\rbrack = 1$, where $\tau_D = \inf\{t > 0 : B_t \notin D\}$, and $(B_t)_{t \ge 0}$ is a Brownian motion. 

 1. Is the domain $\mathbb D \setminus [0, 1)$ regular?
 2. Is every simply connected domain regular?