14. Metrization
• (X, d) : a metric space
Let B = {B(x, ") | x 2 X, " > 0}. Then B is a basis for some topology T
on X. This topology T on X is called the topology generated by B. So,
every metric space is a topological space.
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metric spaces
topological spaces
• (X, T ) : a topological space
(Metrization problem) Is there a metric d on X which generates T ?
(Ans) There is a topological space X which can not be generated by any
metric on d.
(Ex 14.1) X = {a, b}, T = {X, ;}
Then T is a topology on X.
Suppose there is a metric d on X which generates T .
) d(a, b) = r > 0
(* a 6= b)
) B (a, r) = {a} : open in (X, T )
(contradiction)
) Hence T can not be generated by any metric d on X.
(Def ) A topological space (X, T ) is metrizable if there is a metric d on X
which generates T .
1
2
Remark. Let (X, d) be a metric space. Then there is a bounded metric d0
on X which is equivalent to d.
(Thm 14.1) Let {(Xn , dn ) | n 2 N} be a countable collection of metric
1
Y
spaces and X =
be the product space of {(Xn , dn ) | n 2 N}. Then X is
n=1
metrizable.
(pf) By the above remark, we may assume that dn (x, y)  1 for all x, y 2 Xn
and n 2 N. For x = (x1 , x2 , · · · ) and y = (y1 , y2 , · · · ) in X, define
◆2 ! 12
1 ✓
X
dn (xn , yn )
d(x, y) =
n
n=1
) d : well-defined
(* dn (xn , yn )  1 for all xn , yn 2 Xn and n 2 N)
and d : metric on X
Let T be the product topology on X and T 0 be the topology on X generated
by d.
To show : T = T 0
Let U 2 T and x = (x1 , x2 , · · · ) 2 U .
N
\
)x2
pn 1 (B(xn , rn )) ✓ U for some open ball B(xn , rn ) in Xn
n=1
(1  n  N )
(* T : product topoloy)
Let r = min r1 , r22 , · · · , rNN
and y = (y1 , y2 , · · · ) 2 Bd (x, r). Then
◆2 ! 12
1 ✓
X
dn (xn , yn )
d(x, y) =
<r
n
n=1
dn (xn , yn )
 d(x, y) for each n = 1, 2, · · · , N .
n
) dn (xn , yn )  nd(x, y) < nr  rn
Note that
3
) yn 2 B(xn , rn )
N
\
)y2
pn 1 (B(xn , rn ))
n=1
) Bd (x, r) ✓
N
\
n=1
pn 1 (B(xn , rn )) ✓ U
) U : open in (X, T 0 )
Let U 2 T 0 and y = (y1 , y2 · · · ) 2 U .
) y 2 Bd (y, ") ✓ U for some " > 0
Let M be a positive integer with
1
X
n=M +1
(* T 0 : topology induced by d)
1
"2
< .
n2
2
✓ ✓
◆◆
"
Let V =
pn B yn , p
be a basic open set in (X, T ).
2M
n=1
M
\
1
Then for z = (z1 , z2 , · · · ) 2 V ,
◆2
1 ✓
X
d
(y
,
z
)
n
n
n
(d(y, z))2 =
n
n=1
✓
◆2
1
X
dn (yn , zn )

(dn (yn , zn )) +
n
n=1
M
X
M
X
"2
2
n=M +1
"2
"2 " 2
<
+
=
+
= "2
2M
2
2
2
n=1
) z 2 Bd (y, ")
) V ✓ Bd (y, ") ✓ U
) U : open in (X, T )
In all, T = T 0 .
4
(Thm 14.2) (The Urysohn metrization theorem) Every second countable
regular space is metrizable.
(pf) Let X be a second countable regular space.
) X : normal
(by the corollary of Thm 8.9)
Let B = {Bn | n 2 N} be a countable basis for X.
Let I = {(i, j) | Bi ✓ Bj }.
)I=
6 ;
(* X : regular)
) For (i, j) 2 I, Bi and X\Bj : disjoint closed subsets of X
) 9 continuous f : X ! [0, 1] with f (Bi ) = {0} and f (X\Bj ) = {1}
(by Urysohn lemma)
) 9 countable continuous functions {fn | n 2 N}
(* I : countable)
Define a function F : X ! H from X to the Hilbert space H by
✓
◆
f2 (x) f3 (x)
F (x) = f1 (x),
,
,···
2
3
◆2 X
1 ✓
1
X
fn (x)
1
) F : well-defined (*

< 1)
2
n
n
n=1
n=1
To show : F is an embedding.
(1) F : one-to-one
Let x 6= y in X.
) 9 Bj 2 B such that x 2 Bj and y 2
/ Bj
) Bi 2 B such that x 2 Bi ✓ Bi ✓ Bj
(* X : T0 -space)
(* X : regular)
Let fn be the Urysohn function corresponding to (i, j).
) fn (Bi ) = {0} and fn (X\Bj ) = {1}
5
) fn (x) = 0 and fn (y) = 1
) F (x) 6= F (y)
(2) F : continuous
Let x 2 X and B(F (x), ") be an open ball in H.
1
X
1
"2
Let N be a positive integer for which
< .
n2
2
n=N +1
For 1  n  N , 9 open set Vn containing x such that if y 2 Vn ,
fn (y)| < p
|fn (x)
Let V =
N
\
n=1
"
2N
(* fn : continuous at x)
Vn . Then V is an open set containing x and if y 2 V ,
2
(d (F (x), F (y))) =
1 ✓
X
fn (x)
n=1

N
X
n
(fn (x)
fn (y)
n
◆2
2
fn (y)) +
n=1
✓
1
X
fn (x)
n=N +1
n
fn (y)
n
◆2
✓ ◆2
N ✓
1
X
X
"
1
"2
"2
p
<
+
=N·
+
n
2N
2
2N
n=1
n=N +1
"2 "2
+
= "2
2
2
) F (y) 2 B(F (x), ")
=
) F (V ) ✓ B(F (x), ")
) F : continuous at x
(3) F : open map from X to F (X)
Let W be an open set in X and x 2 W .
Claim : F (W ) is open in F (X),
i.e., B F (x), n1 \ F (X) ✓ F (W ) for some n.
◆2
6
B : basis ) x 2 Bj ✓ W for some Bj 2 B
) 9 open set Bi 2 B such that x 2 Bi ✓ Bi ✓ Bj ✓ W
(* X : regular)
Let fn be the Urysohn function corresponding to (i, j).
) fn (Bi ) = {0} and fn (X\Bj ) = {1}
) fn (x) = 0 and fn (X\W ) = {1}
) for any F (y) 2 B F (x), n1 \ F (X), fn (y) 6= 1
(* If fn (y) = 1, then
1 ✓
X
fi (x)
d(F (x), F (y)) =
i
i=1
fn (x)
n
fi (y)
i
◆2 ! 12
fn (y)
1
=
n
n
)
)y2
/ X\W
)y2W
) B F (x), n1 \ F (X) ✓ F (W )
) F (W ) : open in F (X)
(* x : arbitrary)
) F : open map from X to F (X)
Remark. (1) X : a metric space, T.A.E.
(a) X is second countable.
(b) X is separable.
(c) X is Lindelöf.
7
(* (b))(a) separable metric space ) second countable
(a))(c) second countable spaces ) Lindelöf space
(c))(b) Let X be a Lindelöf metric space.
For each k 2 N, B x, k1 | x 2 X is an open cover of X.
) 9 B xi,k , k1 | i 2 N : countable subcover
) D = {xi,k | i, k
1} : countable dense subset of X)
(2) Hence every compact metric space is second countable.
(Thm 14.3) Let X be a compact metric space and Y be a Hausdor↵ space.
If f : X ! Y is a continuous surjection, then Y is metrizable.
(pf) Let X be compact. ) Y : compact
) Y : normal
(* f : continuous)
(* Y : Hausdor↵)
) Y : regular
To show : Y is second countable.
X : second countable
(by above remark)
) 9 countable basis B for X
Let A = {Bi1 [ · · · [ Bin | Bik 2 B, 1  k  n}.
) A : countable
(* B : countable)
For A 2 A, let A⇤ = Y \f (X\A).
) A⇤ : open in Y
(* X\A : closed in X ) X\A : compact
) f (X\A) : compact in Y
8
) f (X\A) : closed in Y
(* Y : Hausdor↵)
) Y \f (X\A) : open in Y )
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A
f
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X\A
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A⇤
f (X\A)
X
Y
Claim : A⇤ = {A⇤ | A 2 A} is a basis for Y .
Let U be an open subset of Y and y 2 U .
) f 1 (y) ✓ f 1 (U )
) f 1 (y) : compact subset of X (* {y} : closed)
n
[
1
) f (y) ✓
Bi ✓ f 1 (U ) for some B1 , · · · , Bn 2 B
i=1
) for A =
n
[
i=1
Bi , y 2 A⇤ ✓ U
((i) If y 2 f (X\A), then f 1 (y) * A
(contradiction to
f 1 (y) ✓ A)
) y 2 Y \f (X\A) = A⇤
(ii) Let f (x) 2 A⇤ , then x 2 A.
(* If x 2 X\A, f (x) 2 f (X\A)
) f (x) 2
/ Y \f (X\A) = A⇤
) f (x) 2 f (A) ✓ f f 1 (U ) ✓ U
) A⇤ ✓ U )
) A⇤ : countable basis for Y
(contradiction to f (x) 2 A⇤ )
9
(Thm 14.4) T.A.E.
(a) X is regular and second countable.
(b) X can be embedded in Hilbert space H.
(c) X is metrizable and separable.
(pf) (a))(b) Urysohn metrization theorem
(b))(c) Suppose X can be embedded in Hilbert space H.
X : metrizable
(* H : metric space).
X : second countable
(* H : second countable).
) X : separable
(c))(a) Suppose X is metrizable and separable.
) X : normal
) X : regular
and X : second countable
(by Thm 5.7)
(Def ) A : a family of subsets of X
A is locally finite if for each x 2 X, there is an open set U with x 2 U such
that U intersects only finitely many members of A. A is
-locally finite
if A is a union of a countable collection of locally finite families. That is,
1
[
A=
An , where An is locally finite for all n 2 N.
n=1
(Thm 14.5) (The Nagata-Smirnov-Bing Metrization Theorem) X is metrizable if and only if X is regular and have a -locally finite basis.