Manifold with boundary - finding the boundary

Question

I have the manifold with boundary $M:= \lbrace (x_1,x_2,x_3) \in \mathbb R^3 : x_1\geq 0, x_1^2+x_2^2+x_3^2=1\rbrace \cup\lbrace (x_1,x_2,x_3) \in \mathbb R^3 : x_1= 0, x_1^2+x_2^2+x_3^2\leq1\rbrace$ and I need to find the boundary of this manifold. I think it is $\lbrace (x_1,x_2,x_3) \in \mathbb R^n : x_1= 0, x_2^2+x_3^2=1\rbrace$, the other option is that the boundary is the empty set? I think the first is right? Am I wrong?

Answer 1

The first set is a hemisphere sitting on the $x_2x_3$-plane, including the boundary of the hemisphere at $x_1=0$. The second set is a disk of radius $1$ in the $x_2x_3$-plane. Can you see that the resulting surface has no boundary?