Graphics Reference
In-Depth Information
Suppose that h :
R
n
Æ
R
m
is a homeomorphism. Define
n
m
H
:
SS
Æ
by
()
=
(
)
()
-
1
H
x
p
oo
h p
x
,
if
x
π
e
,
and
mn
n
+
1
(
)
=
H
e
e
.
n
+
1
m
+
1
The map H will be a homeomorphism and therefore n = m by part (1). The theorem
is proved.
The next three theorems are less trivial.
7.2.3.6. Theorem.
(Invariance of Dimension) If K and L are simplicial complexes
with ΩKΩªΩLΩ, then dim K = dim L.
Proof.
See [AgoM76].
Definition.
The
dimension
of a polyhedron is defined to be the dimension of any
simplicial complex that triangulates it.
Theorem 7.2.3.6 shows the dimension of a polyhedron is a well-defined topologi-
cal invariant.
7.2.3.7. Theorem.
(Invariance of Boundary) If K and L are simplicial complexes
and h : ΩKΩÆΩLΩ is a homeomorphism, then h(Ω∂ KΩ) =Ω∂LΩ.
Proof.
See [AgoM76].
Theorem 7.2.3.7 makes it possible to define the boundary of a polyhedron.
Definition.
Let
X
be a polyhedron. Define the
boundary
of
X
, denoted by ∂
X
, by
(
K,
∂
X
=
j∂
where (K,j) is any triangulation of
X
.
7.2.3.8. Theorem
(Invariance of Domain) If
U
and
V
are homeomorphic subsets of
R
n
and if
U
is open in
R
n
, then so is
V
.
Proof.
See [AgoM76].
Returning to our definition of topological manifolds in Section 5.3, we are finally
able to prove the claimed invariance of two aspects of the definition.
7.2.3.9. Corollary.
The dimension of a topological manifold and its boundary are
well defined.
