Graphics Reference
In-Depth Information
It is easy to show that d* is a metric on
X
with the desired properties (Exercise 5.2.7).
See [Eise74].
Definition.
Let (
X
,d) and (
Y
,d¢) be metric spaces. A one-to-one and onto map
f:
X
Æ
Y
is called an
isometry
between these metric spaces if d¢(f(
p
),f(
q
)) = d(
p
,
q
)
for all
p
,
q
X
. An arbitrary map f :
X
Æ
Y
is called a
local isometry
if for every
p
in
X
there exist d- and d¢-neighborhoods
U
and
V
of
p
and f(
p
), respectively, so that
f|
U
is an isometry between
U
and
V
.
Motions in
R
n
are all isometries.
Definition.
X
. A map f :
X
Æ
Y
is
said to be
(d,d
¢
)-continuous at
p
if for every e>0 there is a d>0 so that d¢(f(
p
),f(
q
))
<efor all
q
Let (
X
,d) and (
Y
,d¢) be metric spaces and let
p
X
with d(
p
,
q
) <d. The map f is said to
(d,d
¢
)-continuous
if it is (d,d¢)-
continuous at every point of
X
.
It is easy to see that in the case of the Euclidean metric this is just the usual def-
inition of continuity of functions f :
R
n
Æ
R
m
. See statement (4.1) in Chapter 4.
Definition.
Let (
X
,d) and (
Y
,d¢) be metric spaces. A map f :
X
Æ
Y
is said to be
(d,d
¢
)-
uniformly continuous
if for every e>0 there is a d>0 so that for all
p
,
q
X
, d(
p
,
q
)
<dimplies that d¢(f(
p
),f(
q
)) <e.
We already pointed out the big difference between uniform and ordinary continuity
in Section 4.2. The d in the uniform continuity case does
not
depend on where we are
and only on the e. Having this independence is often important, and so it is always better
to have a uniformly continuous map rather than just a plain continuous one.
Next, the usual notion of convergence of sequences in
R
n
extends to metric spaces
in a natural way.
Definition.
Let (
X
,d) be a metric space. A sequence of points
p
n
, n = 1,2,...in
X
is
said to
converge to a point
p
in (
X
,d) if for every e>0 there is an m ≥ 1 so that if n ≥
m then d(
p
n
,
p
) <e. The sequence
p
n
is said to
converge
in (
X
,d) if it converges to some
point
p
in (
X
,d).
5.2.6. Theorem.
Let (
X
,d) be a metric space. If a sequence of points
p
n
, n = 1,2,
...in
X
converges to two points
p
and
q
in (
X
,d), then
p
=
q
.
Proof.
Let e>0. By definition, there exists an m so that n > m implies that d(
p
n
,
p
)
<eand d(
p
n
,
q
) <e. But then
(
)
£
(
)
+
(
)
<+=
d
pq
,
d
pp
,
d
p q
,
ee
2
e
,
n
n
which clearly implies that
p
=
q
since e was arbitrary.
Theorem 5.2.6 says that if a sequence converges to a point, then it converges to a
unique
point.
