Graphics Reference
In-Depth Information
Definition.
A
pseudometric
on a set
X
is a function
d:
XX R
¥Æ
satisfying conditions (1), (3), and (4) in the definition of a metric and the following
weakened form of condition (2):
(2¢)d
p
,
p
) = 0 for all
p
X
.
(a)
Show that the relation ~ on
X
defined by
if and only if d
(
)
= 0
pq
~
pq
,
is an equivalence relation on
X
.
(b)
Let
X
* denote the set of equivalence classes of
X
with respect to the relation ~ in
(a). Define
d
** *
:
XX R
¥Æ
by
(
[][]
)
=
(
)
d
*,
pq
d
pq
,
.
Show that d* is a well-defined metric on
X
*.
Section 5.3
Prove that the Euclidean, taxicab, and max metric on
R
n
induce the same topology.
5.3.1.
5.3.2.
Prove equations (5.6) and (5.7).
5.3.3.
Define a homeomorphism between the open interval (0,1) and
R
.
Section 5.4
5.4.1.
Prove that if the maps f
i
:
X
i
Æ
Y
i
are continuous, then so is the map
ff
¥
¥
...
¥
fX X
:
¥
¥
...
¥
X YY
Æ¥
¥
...
¥
Y
.
12
n
1
2
n
1 2
n
We can give a more concrete description of the cone on a subspace of
R
n
. Let
X
R
n
.
5.4.2.
R
n+1
-
R
n
. Prove that the cone C
X
on
X
is homeomorphic to the space
Choose
v
{
(
)
Œ
[]
}
t
x
+-
1
t
v x
Œ
X
and t
0
,
.
Prove that
D
n
ª C
S
n-1
.
5.4.3.
We can give a more concrete description of the suspension of a subspace of
R
n
. Let
X
5.4.4.
R
n+1
-
R
n
and
w
R
n+2
-
R
n+1
. Prove that the suspension
R
n
.
Choose
v
S
X
of
X
is homeomorphic to the space
{
}
(
)
Œ
[]
t
x
+-
1
t
u x
X
,
t
0 1
,
,
and
u
=
v
or
w
.
Prove that
S
n
ª S
S
n-1
.
5.4.5.
