I don't have a clue for question (b) at all. Can I get some help?
Let $A,B \subset S^n$ be disjoint closed subsets. The Smooth Urysohn Theorem guarantees that there is a smooth $\phi: S^n \to [0,1]$ with $\phi|_A \equiv 0$ and $\phi|_B \equiv 1$.
(a) Prove that there is a compact, oriented, codimension-1 submanifold $X$ in $S^n$ that is disjoint from $A$ and $B$ and separates them in the sense that any curve from $A$ to $B$ must intersect $X$.
(b) For any $X$ as in (a), suppose $p$ is in $S^n \subset \mathbb{R}^{n+1}$ with $p, -p \notin X$. Let $S_p$ be the unit $(n-1)$-sphere centered at 0 in the boundary of the half-space in $\mathbb{R}^{n+1}$ for which $p$ is the outward normal. (Note that this definition orients $S_p$.) Let $\pi_p: S^n - \{\pm p\} \to S_p$ be the obvious projection, and let $\delta(p) = \deg(\pi_p|X)$. Prove that $X$ can be constructed as in (a) so that $| \delta{p}|<10$ for all $p$. What are the possible values of $\delta$ in your construction? Prove your answer.