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