Let $ E = \{x \in \mathbb{R}^n: x_i > 0 \text{ for all } i, \sum\limits_{i=1}^n x_i = 1\}$ be the simplex in $\mathbb{R}^n$. If we now define the set
$$E_1 := \{x \in E: x_1 > x_i \text{ for all }i \in \{2,\dots,n\}\}$$
then I am wondering how to check if the set $E\setminus E_1$ is simply connected. I tried picturing things in 3d and I think it is simply connected in that case, but am having trouble finding ways to formally show this for general n. Any hints or direction towards resources would be much appreciated!