Graphics Reference
In-Depth Information
Figure 5.4.
The domain and graph of a func-
tion are homeomorphic.
y
B
2
(p
0
,r)
p
0
= (x
0
,f(x
0
))
f(x)
h
x
x
0
y
p
q
x
Figure 5.5.
A circle is homeomorphic to the boundary of
a square.
ÆÃ
2
h:
RXR
defined by
()
=
(
()
)
hx
xfx
,
is a homeomorphism. To see this, let
p
0
= (x
0
,f(x
0
)) and consider an open ball
B
2
(
p
0
,r)
about
p
0
. See Figure 5.4. Now, B
2
(
p
0
,r) «
X
is an open set of
X
and it is easy to see
that h
-1
(
B
2
(
p
0
,r) «
X
) is an open interval in
R
containing x
0
. This and the fact that h
is obviously a bijection clearly imply that h is a homeomorphism.
The unit circle
S
1
5.3.15. Example.
is homeomorphic to the boundary
X
of the
square [-1,1] ¥ [-1,1]. Define a map
1
h:
XS
Æ
by
p
p
()
=
h
p
.
It is easy to see from Figure 5.5 that h and h
-1
maps open sets to open sets. Open arcs
in
S
1
correspond to open “intervals” in
X
.
