Simplicial Structures in Topology

III.2 Homology of Polyhedra 113
A simplicial function g: K (r) → L is called a simplicial approximation of a map
f : |K| −→|L| if |g|(p) ∈|s( fF(p))| for every p ∈|K (r) | (recall that s( fF(p)) is
the carrier of the point fF(p) and that F : |K (r) |→|K| is the homeomorphism in
Theorem (III.1.4)).
(III.2.4) Theorem (Simplicial Approximation Theorem). Let K =(X,Φ) and
L =(Y,Ψ) be two simplicial complexes and let f : |K|→|L| be a map. Then, there
are an integer r ≥ 0 and a simplicial function g: K (r) → L, which is a simplicial
approximation of f .
Proof. Consider the finite open covering {A(y)|y ∈ Y } of |L| and let ℓ be
the Lebesgue number (see Theorem (I.1.41)) of the finite open covering
{ f −1 (A(y)) | y ∈ Y} of |K|. Letrbe a positive integer such that diam |K (r) |