Let $K$ and $L$ be (finite) simplicial complexes. Explain carefully what is meant by a simplicial approximation to a continuous map $f:|K| \rightarrow|L|$. Indicate briefly how the cartesian product $|K| \times|L|$ may be triangulated.

Two simplicial maps $g, h: K \rightarrow L$ are said to be contiguous if, for each simplex $\sigma$ of $K$, there exists a simplex $\sigma *$ of $L$ such that both $g(\sigma)$ and $h(\sigma)$ are faces of $\sigma *$. Show that:

(i) any two simplicial approximations to a given map $f:|K| \rightarrow|L|$ are contiguous;

(ii) if $g$ and $h$ are contiguous, then they induce homotopic maps $|K| \rightarrow|L|$;

(iii) if $f$ and $g$ are homotopic maps $|K| \rightarrow|L|$, then for some subdivision $K^{(n)}$ of $K$ there exists a sequence $\left(h_{1}, h_{2}, \ldots, h_{m}\right)$ of simplicial maps $K^{(n)} \rightarrow L$ such that $h_{1}$ is a simplicial approximation to $f, h_{m}$ is a simplicial approximation to $g$ and each pair $\left(h_{i}, h_{i+1}\right)$ is contiguous.