= Solution
The <Simplicial approximation theorem> states that a <continuous map> between realizations of finite <simplicial complexes> becomes homotopic to a <simplicial map> after sufficiently many <barycentric subdivisions> of the domain. More precisely, a vertex assignment is a <simplicial approximation> when the image of each open star is contained in the open star of its assigned vertex; sufficiently fine subdivision permits such an assignment. The resulting straight-line <homotopy> stays in the appropriate target simplices.
The vertices of a <barycentric subdivision> are the barycentres of nonempty faces. An $n$-simplex has $2^{n+1}-1$ such faces. A top-dimensional subdivided simplex corresponds to a full flag of faces of sizes $1,2,\ldots,n+1$, equivalently to an ordering of the original vertices. Thus
$$
\boxed{\#\text{vertices}=2^{n+1}-1,\qquad\#\text{top-dimensional simplices}=(n+1)!.}
$$
Triangulate both copies of the sphere using finite complexes. For each fixed subdivision level there are only finitely many vertex maps to the finite target <vertex set>, hence only finitely many <simplicial maps>. Every <continuous map> is homotopic to one of them at some finite level. A <countable union> of <finite sets> is countable, proving \b[at most countably many <homotopy classes>] without needing their classification by degree.
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