Simplicial approximation theorem

Remarks

A vertex map $g$ is a simplicial approximation of $f$ as soon as $f(\operatorname{st} v) \subseteq \operatorname{st} g(v)$ for every vertex $v$, where $\operatorname{st}$ denotes the open star. The open stars of the vertices of $L$ form an open cover of $\abs{L}$, so their preimages under $f$ form an open cover of the compact metric space $\abs{K}$. Repeated barycentric subdivision makes the simplices smaller than a Lebesgue number of this cover. Then each open star of $\operatorname{sd}^m K$ maps into some open star of $L$, and choosing such a vertex for each $v$ gives $g$.

A simplicial approximation $g$ of $f$ is homotopic to $f$ by the straight-line homotopy, which stays in $\operatorname{supp}(f(x))$.

Subdivision is necessary. Let $K = L$ be the boundary of a triangle, so $\abs{K}$ is a circle, and let $f$ wrap this circle twice around itself. A simplicial map $K \to L$ is either a bijection on vertices, of degree $\pm 1$, or misses a vertex, of degree $0$. Since a simplicial approximation is homotopic to $f$, which has degree $2$, no simplicial map $K \to L$ approximates $f$.

Going deeper

For a student

Simplicial approximation shows that the set of homotopy classes of maps $\abs{K} \to \abs{L}$ is countable when $K$ and $L$ are finite. It is used to prove that simplicial homology is a homotopy invariant and agrees with singular homology, and to compute $\pi_k(S^n) = 0$ for $k < n$. The analogue for CW complexes is the cellular approximation theorem.