Graphics Reference
In-Depth Information
(
)
A
=
max
A
,
A
,...,
A
n
.
1
2
Define
(
)
¥
(
)
Æ
n
n
dI
:
RRR
I
by
(
)
=-
dAB
,
A B
.
The map d is a metric and (I(
R
n
),d) is a complete metric space.
18.2.14
Lemma.
The maps
(
)
Æ
(
)
Æ
n
n
n
w I
:
RR
and
mid I
:
RR
are continuous functions with respect to that topology.
Proof.
Easy.
In the future we shall assume that I(
R
n
) is a topological space with the topology
derived from the metric d. It is just the product topology defined from the topology
of I(
R
). Therefore, it makes sense to talk about topological notions such as conver-
gence. Also, we note that the properties we stated earlier for intervals in
R
have
obvious generalizations and hold for intervals in
R
n
. We shall not repeat them here.
Here is one last operation on intervals. It defines the smallest interval that
contains two intervals.
Definition.
If A = [a,b] and B = [c,d] are two intervals in
R
, define
[
(
)
(
)
]
AB
⁄=
min
ac
,
, max
bd
,
.
If A = A
1
¥ A
2
¥ ...¥ A
n
and B = B
1
¥ B
2
¥ ...¥ B
n
are intervals in
R
n
, define
(
)
¥⁄
(
)
¥¥ ⁄
(
)
AB A B
⁄=
⁄
A B
...
A
nn
.
1
1
2
2
18.3
Inclusion Functions
It is convenient to generalize the notion of an induced function.
Let
X
Õ
R
m
and f :
X
Æ
R
n
. An
inclusion function
for f is a function
Definition.
()
Æ
(
)
n
FI
:
XR
I
with the property that for all A ΠI(
X
),
()
Œ
()
f
F A
,
for all x
Œ
A
.
