Graphics Reference
In-Depth Information
Definition.
(The Countability Axioms) A topological space
X
is said to be
first count-
able
if every point of
X
has a countable
local
base. It is said to be
second countable
if it has a countable base.
Second countability is clearly stronger than first countability. For continuity to be
equivalent to sequential continuity one needs first countability. Second countability
is a desirable property that basically means that one can use induction for construc-
tions. It is also related to another concept.
Definition.
A topological space is said to be
separable
if it has a countable dense
subset.
For example,
R
is separable because the rationals that are a countable set are
dense.
5.3.18. Theorem.
A metrizable space is separable if and only if it is second
countable.
Proof.
See [Eise74].
We finish this section with a definition of one of the most important types of nice
topological spaces, namely, manifolds. Manifolds are really the center of attention
of this topic. They are basically spaces that look like Euclidean space locally, but
because we want to allow for manifolds with boundary the definition is slightly
more complicated.
Definition.
A second countable Hausdorff space
M
is called an
n-dimensional topo-
logical manifold
if every point
p
M
has an open neighborhood
V
p
that is homeo-
morphic to an open subset
U
p
of the standard halfplane
R
+
. Let h
p
:
U
p
Æ
V
p
be the
homeomorphism. The
boundary
of
M
, ∂
M
, is defined by
{
}
-
1
n
-
1
()
Œ
∂= Œ
MpM
h
pR
.
p
The points of ∂
M
are called
boundary points
. The set
M
-
∂
M
is called the
interior
of
M
and
its points are called
interior points
. If n is unimportant one calls
M
simply a
topological
manifold
. The dimension of
M
is usually indicated as a superscript and one talks about
the “manifold
M
n
.” A manifold that has no boundary is said to be
closed
.
See Figure 5.6. Euclidean space
R
n
is the archetypical example of an n-dimen-
sional manifold without boundary. Other well-known examples are the open balls
B
n
and the spheres
S
n
. The spheres are closed manifolds. The halfplane
R
+
and the disk
D
n
are the archetypical examples of n-dimensional manifolds with boundary. Their
boundaries are
R
n-1
and the (n - 1)-sphere
S
n-1
, respectively. We shall see many more
examples of manifolds in coming chapters. A simple-minded way of thinking about a
closed two-dimensional manifold is as a space with the property that we can lay a
blanket around every point
p
of the space so that the points of the blanket match is
a one-to-one and onto manner the points of a neighborhood of
p
.
