Graphics Reference
In-Depth Information
5.6.5. Theorem.
(The Intermediate Value Theorem) Let
X
be a topological space
and f :
X
Æ
R
a continuous map. Assume that f(
x
1
) < f(
x
2
) for some
x
1
,
x
2
X
. If c is
a real number so that f(
x
1
) < c < f(
x
2
), then there is an
x
X
with f(
x
) = c.
Proof.
See [Eise74].
Definition.
A connected subset of a topological space
X
that is not properly con-
tained in any connected subset of
X
is called a
component
of
X
.
A more intuitive way to express the notion of component is to say that a compo-
nent is a maximal connected subset.
A simpler notion of connected is:
Definition.
Let
X
be a topological space. We say that
X
is
path-connected
if for any
two points
p
,
q
X
, there is a continuous map f : [0,1] Æ
X
with f(0) =
p
and f(1) =
q
. The map f is called a
path from
p
to
q
. A maximal path-connected subset of a top-
ological space
X
is called a
path-component
of
X
.
5.6.6. Theorem.
Let f :
X
Æ
Y
be a continuous map from a path-connected space
onto a space
Y
. Then
Y
is path-connected.
Proof.
See [Eise74].
5.6.7. Theorem.
A path-connected space is connected.
Proof.
See [Eise74].
Connected does not imply path-connected in general, so that the notion of path-
connected is stronger. For “nice” spaces however these concepts are identical.
5.6.8. Theorem.
A topological manifold is connected if and only if it is path-
connected.
Proof.
See [Eise74].
5.7
Homotopy
We have talked about how topology studies properties of spaces invariant under
deformations (rubber sheet geometry). This section studies deformations of
mappings.
Definition.
Let f, g :
X
Æ
Y
be continuous maps. A
homotopy between f and g
is a
continuous map
¥
[
Æ
h:
X
0,
Y
