Serge Ballif
MATH 527 Homework 2
September 14, 2007
Problem 1.
Consider a map
h
:
R
ω
→
R
ω
,
h
(
x
1
,x
2
,x
3
,...
) = (
x
1
,
4
x
2
,
9
x
3
,...
)
.
(a) Is
h
continuous, when
R
ω
is given by the product topology?
(b) Is
h
continuous, when
R
ω
is given by the box topology?
(c) Is
h
continuous, when
R
ω
is given by the uniform topology?
(a) Yes,
h
is continuous. To show this, we show that the inverse image under
h
of a basis set in
R
ω
is open in
R
ω
. An arbitrary basis set in
R
ω
has the form
U
=
Q
∞
i
=1
U
i
for intervals
U
i
= (
a
i
,b
i
) where the two equations
a
i
=
∞
and
b
i
=
∞
hold except for ﬁnitely many values of
i
. The calculation
h

1
(
{
(
y
1
,y
2
,...,y
i
,...
)
}
) =
{
(
y
1
,
1
4
y
2
,...,
1
i
2
y
i
,...
)
}
shows that
h

1
(
U
) =
Q
∞
i
=1
(
1
i
2
a
i
,
1
i
2
b
i
). This inverse image is also a product of
open intervals of which only ﬁnitely many are not all of
R
. Hence,
h

1
(
U
) is
another basis element of
R
ω
. Thus the inverse image of a basis element is open
under the map
h
. Therefore,
h
is continuous.
(b) Yes,
h
is continuous. The inverse image under
h
of a basis set
U
=
Q
∞
i
=1
(
a
i
,b
i
)
is the basis set (and hence an open set)
h

1
(
U
) =
Q
∞
i
=1
(
1
i
2
a
i
,
1
i
2
b
i
). Hence,
h
is
continuous.
(c) No,
h
is not continuous. To show
h
is not continuous, we will show that the
inverse image of an open ball does not contain an open ball (and hence is not
open). An basis element of
R
ω
in the uniform topology is of the form
B
¯
p
(
x,±
) =
[
δ<±
∞
Y
i
=1
(
x
i

δ,x
i
+
δ
)
.
The property