Simplicial Structures in Topology

II.2 Abstract Simplicial Complexes 53
(iii): Let ℓ be a ray with origin p, andletq be the point of ℓ determined by the
condition
d(p,q)=sup{d(p,q ′ ) | q ′ ∈ ℓ ∩ D(p)}.
Then, |s(q)| ∈S(p); otherwise, we could extend ℓ in D(p) beyond q and thus we
would have q ∈ S(p). On the other hand, because s(q) is a face of a simplex containing
p, the vertices of s(p) and s(q) define a simplex of which s(p) is a face (in fact,
s(p) ∪ s(q) is a simplex of the simplicial complex D(p)). The points of ℓ beyond q
cannot be in S(p) and the open segment (p,q) is contained in D(p) � S(p). Hence,
ℓ intersects S(p) in one point only. �
Notice that D(p) is not necessarily convex; at any rate, as we have seen in part
(ii) of the previous theorem, D(p) is endowed with a certain kind of convexity in
the sense that, for every q ∈ D(p), the segment [p,q] is entirely contained in D(p).
We say that D(p) is p-convex (star convex). Theorem (II.2.10) allows us to define
amap
πp : D(p) � {p}→S(p) , q ↦→ ℓp,q ∩ S(p)
where ℓp,q is the ray with origin p and containing q; the function πp is the radial
projection with center p from D(p) onto S(p). Let i: S(p) → D(p) � {p} be the
inclusion map; then πpi = 1 S(p),andiπp is homotopic to the identity map of D(p)�
{p} onto itself with homotopy given by the map
H : (D(p) � {p}) × I → D(p) � {p} , (q,t) ↦→ (1 − t)q + tπp(q).
Hence, S(p) is a deformation retract of D(p) � {p} (see Exercise 2, Sect. I.2).
This shows another similarity between the spaces D(p), S(p) and, respectively, the
n-dimensional Euclidean disk and its boundary.
The next result (cf. [24]) will be used only when studying triangulable manifolds
(Sect. V.1); the reader could thus leave it for later on.
(II.2.11) Theorem. Let f : |K|→|L| be a homeomorphism. Then, for every p ∈|K|,
S(p) and S( f (p)) are of the same homotopy type.
Proof. Assume that s( f (p)) = {y0,...,yn} and let U = |s( f (p))| � | s( f (p))| be the
interior of |s( f (p))|, that is to say, the set of all q ∈|L| such that q(yi) > 0, for
every i = 0,...,n. Notice that U is an open set of D( f (p)); moreover, f −1 (U) is
an open set of |K| containing p. The bounded, compact set D(p) can be shrunk at
will: in fact, for any real number 0 < λ ≤ 1wedefinethecompression λD(p) as
the set of all points r =(1− λ )p + λ q, foreveryq∈D(p); observe that λD(p)
is a closed subset of |K|, and is homeomorphic to D(p). Let λ ∈ (0,1] be such
that
p ∈ λD(p) ⊂ f −1 (U);
then
f (p) ∈ f (λD(p)) ⊂ U ⊂ D( f (p)).
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