Graphics Reference
In-Depth Information
Definition.
Let f :
X
Æ
Y
be a map from a topological space
X
to a topological space
Y
. Let
p
X
. The map f is said to be
continuous at
p
if for every neighborhood
N
of f(
p
) in
Y
the set f
-1
(
N
) is a neighborhood of
p
in
X
. The map f is said to be
continuous
if it is continuous at every point of
X
.
5.3.9. Theorem.
Let f :
X
Æ
Y
be a map from a topological space
X
to a topologi-
cal space
Y
. The map f is continuous if and only if f
-1
(
V
) is open in
X
for all open sets
V
in
Y
.
Proof.
Easy.
We already pointed out in Section 4.2 that in the case of Euclidean spaces the def-
inition of continuity agrees with the standard epsilon-delta definition from calculus.
The messy epsilons and deltas obscure the issues that are really at stake. Of course,
that definition would actually not be possible here anyway since we do not have a
metric. Continuous maps are the natural maps for topological spaces because they
involve the only thing that we have in the context of a topology on a set, namely, open
sets. Note, however, that we are
not
saying that a continuous map sends open sets to
open sets. Such a definition might seem like the obvious one at first glance, but it
would not capture what we have in mind.
Definition.
A map f :
X
Æ
Y
from a topological space
X
to a topological space
Y
is
said to be an
open map
if it maps every open set of
X
to an open set of
Y
. It is said
to be a
closed map
if it maps every closed set of
X
to an closed set of
Y
.
The following example shows the difference between a map being continuous,
open, and/or closed:
5.3.10. Example.
Let
X
be the reals
R
with the standard topology and let
Y
be the
reals with the discrete topology. Then the identity map from
X
to
Y
is not continuous
but both open and closed. On the other hand, the identity map from
Y
to
X
is con-
tinuous but neither open nor closed.
5.3.11. Theorem.
If f :
X
Æ
Y
and g :
Y
Æ
Z
are continuous maps between topolog-
ical spaces, then the composite map h = fg:
X
Æ
Z
is a continuous map.
o
Proof.
This follows easily from the definitions.
The next theorem shows that we can piece together continuous maps to get a
global continuous map.
5.3.12. Theorem.
Let {
A
i
}
i
I
be a covering of a topological space
X
by subspaces
with the property that either all the
A
i
are open or all the
A
i
are closed and I is finite.
If f :
X
Æ
Y
is a map to another topological space, then f is continuous if the restric-
tion maps f |
A
i
are continuous for each i
I.
Proof.
Suppose that the
A
i
are open. Let
V
be an open set in
Y
. If f |
A
i
is continu-
ous, then
