Simplicial Structures in Topology

II.2 Abstract Simplicial Complexes 49
simplicial complex K ′ ⊂ R n associated with K. We shall see in a short while that
K ′ is isomorphic to the geometric realization |K| (actually, there exists an isometry
between these two metric spaces; furthermore, the set of all functions X → R≥0
coincides with the positive quadrant of R n ).
The following statement holds true: two points p,q ∈|K| coincide if and only if
they have the same barycentric coordinates.
The geometric realization functor
| |: Csim −→ Top
is defined over an object K ∈ Csim as the geometric realization |K|, and over a
morphism f ∈ Csim(K,L) as
| f |: |K|→|L|, | f |(∑αixi)=∑αi f (xi) .
To prove that ||is indeed a functor, we need the following result.
(II.2.7) Theorem. The function | f | induced from a simplicial function f : K → Lis
continuous.
Proof. It is enough to prove that, for every p ∈|K|, there exists a constant c(p) > 0
which depends on p and such that, for every q ∈|K|, d(| f |(p),| f |(q)) ≤ c(p)d(p,q).
Assume that
s(p)={x0,...,xn} and s(q)={y0,...,ym}
and also that p(xi)=αi for i = 0,...,n,andq(y j)=β j for j = 0,...,m. We consider
three cases.
Case 1: s(p) ∩ s(q)=/0 - In this situation
d(p,q)=
� n
∑ α
i=0
2 i +
m
∑
j=0
β 2 j ≥
�
n
∑ α
i=0
2 i ;
because ∑ n i=0 αi = 1, ∑ n i=0 α2 i has its minimum value only when αi = 1/(n + 1),for
every i = 0,...,n. It follows that d(p,q) ≥ 1/ √ n + 1and
d(| f |(p),| f |(q))
d(p,q)
(recall that d(| f |(p),| f |(q)) ≤ √ 2); so,
≤
√ 2
1/ √ n + 1
d(| f |(p),| f |(q)) ≤ � 2(n + 1)d(p,q);
thus, we define c(p)= � 2(n + 1).
Case 2: s(p) ∩ s(q) �= /0, but s(p) �⊂ s(q) and s(q) �⊂ s(p) –