In "An Introduction to Algebraic Topology" of Rotman, Exercise 1.31 asks to show that the equator $\mathbb{S}^{n-1}$ is a deformation retract of $\mathbb{S}^n\setminus\{a, b\}$.
I thought that if one thinks to the usual sphere in the space, then one just enlarge the holes to flatten $\mathbb{S}^3$ onto its equator, but I cannot understand how to prove this in general for $n$, probably it's the same idea but I cannot construct the required function.