Econ 204
Corrections to de la Fuente
Note that Item 10 was added 7/22/09
1. On page 23, de la Fuente presents two definitions of correspondence.
In the second definition, de la Fuente requires that for all
x
∈
X
,
Ψ(
x
) =
∅
. The first definition simply says that Ψ is a function from
X
to 2
Y
, the collection of all subsets of
Y
, and appears to believe that
this implies that Ψ(
x
) =
∅
; since
∅ ∈
2
Y
, this belief is not correct. In
the first definition, he should have said “a correspondence is a function
from
X
to 2
Y
such that for all
x
∈
X
, Ψ(
x
) =
∅
.”
2. Theorem 5.2, page 64, should read as follows:
Theorem 1 (5.2’)
Let
(
X, d
)
and
(
Y, ρ
)
be two metric spaces,
A
⊆
X
,
f
:
A
→
Y
, and
x
0
a limit point of
A
. Then
f
has limit
y
0
as
x
→
x
0
if
and only if for every sequence
{
x
n
}
that converges to
x
0
in
(
X, d
)
with
x
n
∈
A
for every
n
and
x
n
=
x
0
, the sequence
{
f
(
x
n
)
}
converges to
y
0
in
(
Y, ρ
)
.
Comment: As stated in de la Fuente, the metric space (
X, d
) is the
ambient space, so every limit point of
X
must be an element of
X
;
there is nothing outside of
X
to which a sequence in
X
can converge.
Thus, as stated in de la Fuente, we must have
x
0
∈
X
and thus,
x
0
must be in the domain of
f
. The revised statement just given allows
x
0
to lie outside the domain of
f
.
3. De la Fuente uses a weaker definition of homeomorphism (Definition
6.20, page 74) than most texts; usually, a homeomorphism is required to
be a surjection. For example, the injection map
I
: [0
,
1]
→
R
defined
by
I
(
x
) =
x
would not be called a homeomorphism in most texts
because it is not onto, but it is a homeomorphism under de la Fuente’s
weaker definition. This creates trouble in Theorem 6.21(ii). [0
,
1] is an
open set in the metric space [0
,
1], but its image
I
([0
,
1]) = [0
,
1] is
not
an open set in
R
, so Theorem 6.21 is false as stated. Theorem 6.21 is
true if we assume that
f
: (
X, d
)
→
(
Y, ρ
) is one-to-one
and onto
, or if
1