Simplicial Structures in Topology

II.2 Abstract Simplicial Complexes 51
d(| f |(p),| f |(q)) ≤ � 2(n + 1)d(p,q).
Case 3: Let us assume that s(p) ⊂ s(q). Rewrite the indices of the elements of s(p)
and s(q) in such a way that, xi = yi for every i = 0,...,n. Similar to the previous
case, we consider the elements
�
xi = yi, 0 ≤ i ≤ n
zi =
y j, n + 1 ≤ j ≤ m
and the real numbers
�
−αi + βi, 0 ≤ i ≤ n
γi =
β j, n + 1 ≤ j ≤ m.
If −αi +βi ≥ 0foreveryi = 0,...,n,thens(p)=s(q) and p = q, because ∑ n i=0 αi =
1. Hence, there exists a number 0 ≤ i ≤ n such that −αi + βi < 0. At this point, we
argue as in the previous case. If s(q) ⊂ s(p), we use an analogous procedure. �
In particular, the following result holds true:
(II.2.8) Theorem. Any piecewise linear function (the simplicial realization of a
simplicial function)
�
is continuous.
F : |K|→|L| , F
�
n
∑ αixi
i=0
=
n
∑
i=0
αiF(xi)
Hence | | is a functor.
We now investigate some of the properties of the geometric realization of a simplicial
complex. Recall that it is possible to characterize a convex set X of an
Euclidean space as follows: for every p,q ∈ X, the segment [p,q], with end-points p
and q, is contained in X. As we are going to see in the next theorem, this convexity
property is valid for the geometric realization of the complex σ (called geometric
simplex), for every simplex σ of a simplicial complex K.
(II.2.9) Theorem. Let K =(X,Φ) be a simplicial complex. The following results
hold true:
(i) The geometric realization σ of any simplex σ ∈ Φ is convex.
(ii) For every two simplexes σ,τ ∈ Φ we have
(iii) For every σ ∈ Φ, σ is compact .
|σ|∩|τ| = |σ ∩ τ|.
Proof. (i) Assume that σ = {x0,...,xn} and let p,q be arbitrary points of |σ|; suppose
that p = ∑ n i=0 αixi and q = ∑ n i=0 βixi. The segment [p,q] is the set of all points
r = tp+(1 −t)q,foreveryt ∈ [0,1]. Then