Simplicial Structures in Topology

V.1 Topological Manifolds 173
is a bijection. By Lemma (I.3.2), the projection q (and so also its restriction q|U)
is a map both open and closed; therefore, q|U is a homeomorphism. Since U, as
an open set of X, isann-manifold, (see Lemma (V.1.1)), it follows that the point
[x] ∈ X/G belongs to an open set of X/G homeomorphic to an open set of R n .To
reach the conclusion that X/G is an n-manifold, we need to demonstrate that X/G
is a Hausdorff space. But this is a direct consequence of Theorem (I.1.29), because
of the hypotheses. �
Since the real projective space may be written as the quotient RP n = S n /Z2,itis
a consequence of this theorem that RP n is an n-manifold.
V.1.1 Triangulable Manifolds
An n-manifold X is triangulable if there exists a polyhedron |K| homeomorphic
to X. The real projective space RP n is an example of a triangulable manifold (see
Sect. III.5). An open disk ˚D n r(x) ⊂ R n with radius r and center x ∈ R n is triangulable.
Fig. V.1 Triangulated torus
Fig. V.2 Triangulated torus
(with fewer triangles)
We now prove that the dimension of the simplicial complex K =(X,Φ) equals
the dimension of the manifold X; moreover, if X is connected, the spaces |σ|⊂|K|
appear in a particular way, in other words, each two of them either intersect each