1st December 2004
Munkres
§
35
Ex. 35.3.
Let
X
be a metrizable topological space.
(i)
⇒
(ii): (We prove the contrapositive.)
Let
d
be any metric on
X
and
ϕ
:
X
→
R
be an
unbounded realvalued function on
X
. Then
d
(
x, y
) =
d
(
x, y
) +

ϕ
(
x
)

ϕ
(
y
)

is an unbounded
metric on
X
that induces the same topology as
d
since
B
d
(
x, ε
)
⊂
B
d
(
x, ε
)
⊂
B
d
(
x, δ
)
for any
ε >
0 and any
δ >
0 such that
δ <
1
2
ε
and
d
(
x, y
)
< δ
⇒ 
ϕ
(
x
)

ϕ
(
y
)

<
1
2
ε
.
(ii)
⇒
(iii): (We prove the contrapositive.) Let
X
be a normal space that is not limit point
compact.
Then there exists a closed infinite subset
A
⊂
X
[Thm 17.6].
Let
f
:
X
→
R
be the
extension [Thm 35.1] of any surjection
A
→
Z
+
. Then
f
is unbounded.
(iii)
⇒
(i): Any limit point compact metrizable space is compact [Thm 28.2]; any metric on
X
is continuous [Ex 20.3], hence bounded [Thm 26.5].
Ex. 35.4.
Let
Z
be a topological space and
Y
⊂
Z
a subspace.
Y
is a retract of
Z
if the identity
map on
Y
extends continuously to
Z
, i.e. if there exists a continuous map
r
:
Z
→
Y
such that
Y
Y
Z
r
commutes.
(a)
.
Y
=
{
z
∈
Z

r
(
z
) =
z
}
is closed if
Z
is Hausdorff [Ex. 31.5].
(b)
.
Any retract of
R
2
is connected [Thm 23.5] but
A
is not connected.
(c)
.
The continuous map
r
(
x
) =
x/

x

is a retraction of the punctured plane
R
2
 {
0
}
onto the
circle
S
1
⊂
R
2
 {
0
}
.
Ex. 35.5.
A space
Y
has the UEP if the diagram
(1)
A
f
Y
X
f
has a solution for any closed subspace
A
of a normal space
X
.
(a)
.
Another way of formulating the Tietze extension theorem [Thm 35.1] is: [0
,
1], [0
,
1), and
(0
,
1)
R
have the UEP. By the universal property of product spaces [Thm 19.6], map(
A,
X
α
) =
map(
A, X
α
), any product of spaces with the UEP has the UEP.
(b)
.
Any retract
Y
of a UEP space
Z
is a UEP space for in the situation
A
f
Y
Z
r
X
f
the continuous map
r
f
:
X
→
Y
extends
f
:
A
→
Y
.
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 Fall '08
 Brown
 Logic, Topology, continuous map, Normal space, UEP, closed subspace, absolute retract

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