Simplicial Structures in Topology

56 II Simplicial Complexes
(not necessarily finite) and Φ is a set of nonempty, finite subsets of X satisfying the
following properties:
1. (∀x ∈ X) , {x}∈Φ ,
2. (∀σ ∈ Φ)(∀σ ′ ⊂ σ , σ ′ �= /0) , σ ′ ∈ Φ.
The price we must pay is a strengthening of the topology of K. We keep the metric
d : |K|×|K|−→R≥0
�
(∀p,q ∈|K|) d(p,q)= ∑ (p(x) − q(x))
x∈X
2
which defines a topology on K. While the necessary condition of Theorem (II.2.12)
is still valid, the sufficient condition does not hold because, to prove it, we need
the assumption that X is finite. However, it is precisely the topology of Theorem
(II.2.12) that we impose on |K|; in other words, we must exchange the metric
topology of K with a finer topology. We say that
F ⊂|K| is closed ⇐⇒ (∀σ ∈ Φ) F ∩|σ| is closed in σ .
This topology is normally called “weak topology”; this is somehow a strange
name, considering the fact that the weak topology for K is finer (that is to say, has
more open sets) than the metric topology.
II.2.2 Simplicial Complexes and Immersions
We have proved, aided by the geometric realization functor, that every abstract finite
simplicial complex K can be immersed in an Euclidean space and hence can be
viewed as an Euclidean simplicial complex. The dimension of the Euclidean space
in question is equal to the number of vertices, say m, of the complex. At this point,
we ask ourselves whether it is possible to immerse K in an Euclidean space of
dimension lower than m. The next theorem answers that question. Before stating the
theorem, we define Euclidean simplicial complexes in a different (but equivalent)
fashion. Let K ⊂ R n be a union of finitely many Euclidean simplexes of R n such
that
1. If σ ⊂ K, every face of σ is in K.
2. The intersection of any two Euclidean simplexes of K is a face of both.
It is not difficult to prove that a set of simplexes of R n verifying the previous
conditions is an Euclidean simplicial complex as defined in Sect. II.1. We also notice
that if F ⊂ K is closed, the intersection F ∩σ is closed in σ for every Euclidean
simplex σ of K; conversely,ifF is a subset of K such that, for every Euclidean
simplex σ of K, F ∩ σ is closed in σ, thenF is closed in K because F is the finite
union of the closed sets F ∩ σ. Clearly, an Euclidean complex K of R n is
compact and closed in R n . We now state the immersion theorem for simplicial
complexes.
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