Graphics Reference
In-Depth Information
Figure 5.6.
Neighborhoods
of boundary and interior
points of a manifold.
R
n
h
q
U
q
M
n
V
q
q
h
p
V
p
R
n
p
+
U
p
∂M
R
n- 1
h
p
-1
(p)
We shall show later (Corollary 7.2.3.9) that both the dimension and boundary of
a manifold are well defined and do not depend on the neighborhoods
V
p
or the home-
omorphisms h
p
. Boundary points are clearly different from interior points. This is
easy to see in the one-dimensional case. Consider the interval
I
= [0,1], which is a one-
dimensional manifold with boundary. Removing a boundary point such as 0 does not
disconnect the space, but removing any interior point would. The following facts are
easily proved:
(1) Every point of a manifold without boundary has a neighborhood homeomor-
phic to
R
n
.
(2) The boundary of an n-dimensional manifold is an (n - 1)-dimensional mani-
fold without boundary.
Because there are other types of manifolds (Chapter 8 will introduce
diffentiable
manifolds and there are also
piecewise linear
or
PL
manifolds) we shall often drop the
adjective “topological” and simply refer to a “manifold.” The context will always deter-
mine the type if it is important.
Some definitions of topological manifolds do not require second countability.
The reason for requiring a manifold to be second countable in this topic is a
practical one. Without it we would lose some properties of manifolds, such as
metrizability, and many important results in differential topology described in Chapter
8, such as the Whitney imbedding theorem and the Sard theorem, would no longer
hold.
Definition.
A (
topological
)
surface
is a two-dimensional topological manifold. A
(
topological
)
curve
is a one-dimensional topological manifold.
5.4
Constructing New Topological Spaces
This section describes some standard construction with which one can define new
topological spaces from old ones. The first of these is the important concept of quo-
tient spaces.
