Graphics Reference
In-Depth Information
Figure 5.3.
Piecing together continuous functions.
y
1
x
-1
1
2
-
1
-
1
()
«=
(
)
(
)
f
VA
f
A V
i
i
is open in
A
i
and
X
. Therefore,
UI
(
)
-
1
()
=
-
1
()
«=
()
«=
-
1
-
1
()
«
f
V
f
V
XV
f
A
f
V
A
i
i
iI
Œ
iI
Œ
is open in
X
, which proves the continuity of f. The proof of the theorem in the case
where the
A
i
are closed is also an easy consequence of the definitions. See [Eise74].
5.3.13. Example.
By Theorem 5.3.12 the map f : [-1,2] Æ
R
defined by
2
()
=-
[
]
fx
x
+
1
,
if x
Œ-
1 0
,
,
Œ
[]
=-
1
x
,
if
x
0 1
,
,
2
Œ
[]
=-
x
+
32
x
-
,
if
x
12
,
,
is a continuous map. See Figure 5.3.
Definition.
Let
X
and
Y
be topological spaces. A bijection f :
X
Æ
Y
is called a
home-
omorphism
if and only if both f and f
-1
are continuous maps. Two spaces
X
and
Y
are
said to be
homeomorphic
, and we write
X
ª
Y
if there exists a homeomorphism
f:
X
Æ
Y
. Any property of a space that is preserved by a homeomorphism is called a
topological invariant
.
Homeomorphisms capture the natural notion of equivalence between topological
spaces. The identity map of a topological space is a homeomorphism. Homeomor-
phisms are both open and closed maps. To find a homeomorphism between spaces
one first has to find a bijection between them and then needs to show that it preserves
open (or closed) sets.
5.3.14. Example.
Let f :
R
Æ
R
be a continuous map and let
{
(
()
)
}
X
=
xf
,
x
x
Œ
R
be the graph of f with the induced topology from
R
2
. Then the map
