Graphics Reference
In-Depth Information
where we assume that the point
does not belong
X
. Then the set
{
}
T
=»
T
XKK
-
is a compact subset of
X
•
•
is a topology for
X
into a compact topological space containing
X
as
a dense subspace. The space
X
is unique in the sense that if
Y
is any compact Haus-
dorff space that has a point
p
, so that
Y
- {
p
} is homeomorphic to
X
, then this home-
omorphism extends to a homeomorphism from
Y
onto
X
, which sends
p
to
that makes
X
.
Proof.
See [Eise74].
Definition.
The topological space (
X
, T
) is called the
one-point compactification
of
X
and the point
is called the
point at infinity
in
X
.
The obvious example of a one-point compactification is
S
n
, which is the one-point
compactification of
R
n
.
5.6
Connectedness
Connectedness can be defined in a number of ways. Except for the fact that we are
dealing with topological spaces, some of the definitions here will be the same as those
in Section 4.2, but we shall begin with the pure topological notion.
Definition.
A topological space
X
is said to be
connected
if
X
cannot be written in
the form
A
»
B
, where
A
and
B
are two
nonempty
disjoint open subsets of
X
.
5.6.1. Theorem.
Let f :
X
Æ
Y
be a continuous map between topological spaces
X
and
Y
. If
X
is connected, then so is f(
X
).
Proof.
See [Eise74].
5.6.2. Corollary.
Connectedness is a topological property.
5.6.3. Theorem.
Consider a collection of nonempty spaces
X
i
, 1 £ i £ k. All the
X
i
are connected if and only if the product
X
1
¥
X
2
¥ ...¥
X
k
is connected.
Proof.
See [Eise74].
R
n
is connected.
5.6.4. Theorem.
Proof.
One first proves that
R
is connected and then uses Theorem 5.6.3. See
[Eise74].
The next theorem uses the connectedness of
R
and generalizes the usual inter-
mediate value theorem learned in calculus.
