Simplicial Structures in Topology

III.1 The Category of Polyhedra 105
(III.1.5) Theorem. For every real number ε > 0, there is a positive integer r such
that diam |K (r) | < ε.
Proof. Suppose that dimK = n and take a 1-simplex {σ0,σ1} of K (1) arbitrarily; we
may assume that
and
σ0 = {xi 0 ,...,xiq }
σ1 = {xi 0 ,...,xiq ,xj 0 ,...,x jp }
where p + q + 1 ≤ n. Since|K| ∼ = |K (1) | (actually, it is useful to make the identification
|K|≡|K (1) |), the length of the 1-simplex of |K (1) | represented abstractly by
{σ0,σ1} is computed through the formula
because
=
Therefore,
d
� q
∑
k=0
� � 1
q + 1 −
1
q + 1 xi , k
q
∑
k=0
1
p + q + 2 xi + k
�2 1
(q + 1)+
p + q + 2
�
p
∑
ℓ=0
1
p + q + 2 x �
jℓ =
1
p + q + 2
�
1 (p + 1)(p + q + 2)
=
=
p + q + 2 q + 1
= p + q + 1
�
(p + 1)(p + q + 2)
p + q + 2 (p + q + 1) 2 n √
< 2
(q + 1) n + 1
p + q + 1 n (p + 1)(p + q + 2)
< ,
p + q + 2 n + 1 (p + q + 1) 2 < 2 .
(q + 1)
�2 �1/2 (p + 1) =
diam |K (1) | < n √
2 . �
n + 1
Note that the dimension of a simplicial complex is not affected by successive
barycentric subdivisions. Besides, by definition, the dimension of a polyhedron |K|
is the dimension of the simplicial complex K,thatistosay,dim|K| = dimK.
Let L be a simplicial subcomplex of K; the pair of polyhedra (|K|,|L|) has the
Homotopy Extension Property: for every topological space Z and every pair of
maps
f : |K|×{0}−→Z
G: |L|×I −→ Z
with the same restriction to |L|×{0}, there is a map (not necessarily unique)
F : |K|×I → Z whose restrictions to |K|×{0} and |L|×I coincide with f and
G (see the following commutative diagram).