Q: Boundary of a part of the sphere I am asking myself what is the boundary of the set
$$
M:=\{\vec{x}=(x_1,x_2,\ldots,x_n)\in S^{n-1}: x_i>0\textrm{ for all }1\leq i\leq n\}?
$$
(Here $S^{n-1}:=\{\vec{x}\in\mathbb{R}^n: \lVert\vec{x}\rVert=1\}$ denotes the sphere.)
I think the boundary $\partial M$ of $M$ consists of those $\vec{x}=(x_1,x_2,\ldots,x_n)\in S^{n-1}$ with at least one and at most $n-1$ entries $0$ and the remaining entries positive.
Example: $n=3$. I think
$$
\partial M=A\cup B\cup C\cup D,
$$
where
$$
A:=\left\{(1,0,0),(0,1,0),(0,0,1)\right\}
$$
$$
B:=\left\{(x_1,x_2,0): x_1,x_2>0\textrm{ and }x_1^2+x_2^2=1\right\}
$$
$$
C:=\left\{(x_1,0,x_3): x_1,x_3>0\textrm{ and }x_1^2+x_3^2=1\right\}
$$
$$
D:=\left\{(0,x_2,x_3): x_2,x_3>0\textrm{ and }x_2^2+x_3^2=1\right\}
$$

A: This is the definition of Boundary point taken below,
$$\textbf{x} \in S \text{ is boundary point } \leftrightarrow \forall \epsilon >0 \hspace{0.2cm}  \Big( V_{\epsilon}(\textbf{x}) \cap S \neq \phi \Big) \land \Big( V_{\epsilon}(\textbf{x}) \cap \bar{S}  \neq \phi \Big) $$ where $V_{\epsilon}(\textbf{x})$ is $\epsilon$ neigborhood around $
\textbf{x}$
then if  $ M = \{ \textbf{x} \in \mathbb{R}^n : ||\textbf{x}|| = 1 , x_i >0 \}$ then its boundary is following 
$$\partial M = \{ \textbf{x} \in \mathbb{R}^n : ||\textbf{x}|| = 1 , x_i \geq 0 \}$$
