Simplicial Structures in Topology

Chapter V
Triangulable Manifolds
V.1 Topological Manifolds
A Hausdorff topological space X is called an n-dimensional manifold 1 or simply an
n-manifold, if for every point x ∈ X there exists an open set U of X that contains x
and is homeomorphic to an open set of R n . Hence, an n-manifold X is characterized
by a set A = {(Ui,φi) | i ∈ J}, whereUi are open sets covering X, andφi is a
homeomorphism from Ui onto an open set of R n .ThesetA is the atlas of X and
each pair (Ui,φi) is a chart of X.
Even before we give some examples of manifolds, we note that the condition that
X be Hausdorff is an integral part of the definition and does not depend on the other
conditions. In fact, consider
X =] − 1,2]={x ∈ R|−1 < x ≤ 2}
with the topology given by the set U of open sets U,whereU∈Uifand only if one
of the following conditions holds true: (a) U = X; (b)U = /0; (c) U is any union of
sets such as ]α,β[ with −1 ≤ α < β ≤ 2or]α,0[∪]β,2], where−1≤α < 0and
−1 ≤ β < 2. This is not a Hausdorff space since any open set containing 0 intersects,
any open set containing 2. On the other hand, any x ∈ X � {2} is contained by an
open set homeomorphic to an open set of R; regarding the point x = 2, the reader
may verify that the open set U =]− 1 2 ,0[∪] 3 2 ,2] is homeomorphic to an open interval
of R; hence, X has the properties of 1-manifold except for the Hausdorff separation
property. Here is a simple and useful result:
(V.1.1) Lemma. Let V be an open set in an n-manifold X. Then V is an n-manifold.
Proof. For any x ∈ V ⊂ X, let(U,φ) be a chart of X containing x. Then U ∩ V
is open in V containing x and φ(U ∩ V ) is an open set of R n homeomorphic to
U ∩V. �
1 In the literature, it is sometimes required that the topological space X fulfill other conditions to
be defined as a manifold (e.g., second countable, paracompact).
D.L. Ferrario and R.A. Piccinini, Simplicial Structures in Topology, 171
CMS Books in Mathematics, DOI 10.1007/978-1-4419-7236-1 V,
© Springer Science+Business Media, LLC 2011