Nagata, Jun-iti

Osaka City University Repository

Jou rn al of the In stitu te  of Polytechnics 
O saka C ity  U n iversity ,
Vol. 8. No. 2 , Series A
A contribution to the theory of metrization
By Jun-iti N a g a t a
(Received M arch 18, 1957)
Since Alexandroff and Urysohn's work various theorems concerning metriz- 
ability of a toplogical space were gotten by many mathematicians. Their 
methods of proofs, however, are generally various and rather complicated. The 
purpose of this paper is to give a systematic account of the theory of metriza­
tion. A main theorem on the metrizability of a T^-space will be proved first, 
and then it will be shown that this theorem contains a large number of metriza­
tion theorems as direct consequences.
To prove our main theorem we use the following theorem due to E. Michael^  ^
as well as the well-known theorem of P. Alexandroff and P. Urysohn.
M ichael’s theorem. A regular topological space R is paracompact i f  and only
i f  every open covering o f  R has an open refinement y  55^ *, where each is a
i=\
locally finite collection o f open subsets o f R.
T h e o re m  I . In  order that a Ti-topological space R is metrizable it is neces­
sary and sufficient that one can assign a nbd [^neighborhood) basis {U^ [^%)\n =  I,
2 , ---- }  fo r  every point x o f R such that for every n and each point % o f R there
exist nbds Sl^ {x), Sl{x) o f % satisfying
i) y<^Un {x) implies S i (y)  ^ (x) =  <j>,
ii) 3» e  5^  {x) implies (y)
Proof. Since the necessity is clear, we prove only the sufficiency. To begin 
with, R is regular because the condition i) of this proposition implies Sl(x) c  
Un{ )^. Next, to show that R is paracompact, we take an arbitrary open cover­
ing ^={Fo.|a<T} of R^K If we let
 ^ [U,„{x)
^ [S\{x) \U„{X)CZV„,,},
=  K w  =  1 , 2 ,  . . . . . a < T ) ,
then for each m , n i s   ^ locally finite open collection.
To show the local finiteness of we choose, for an arbitrary point p  of
The content of this paper is the datail of our brief note in P ro c . of Jap an  Acad , vol. 33, 
I) See [4]. 2) ii) can ’t  be omitted, ^ ^ i ^ , U n ( p ) = - { q \  d[p, q)
< \ / n } i o x p ^ ( x , y )  with5^>0, U , y' )  \y' < n - { p}  l or  
P ^ (x, 0). Then H with the nbd basis { U^( p } }  is a non- naetrizable T^ -space satisfying i). 
3) In  this proof we denote by ca, y, ordinal numbers and by A° the interior of A .
R, OC ( S r )  such that p^V^i^n,a> P^Vn',,n,  ^ (/?<a). Then it follows from the condi- 
tion i) that S l(p ) r, 4> (;5<a), which implies Sl,(p) ^ (/5<oc). If
a<T, then since p^Vjn,n,(^^yn,a is open, we obtain a nbd of P
satisfying nMnt,n,y==<j> (y>a). Therefore the nbd Sl^(p) nVn,a  of p  intersects 
at most one of elements of Tlm,n> proving the local finiteness of
To assert that S   ^ 9)1^ ,^  covers R, we consider an arbitrary point p  
of R. Let P ^ V a, P ^ V ^  (j8<oc), a < r , then we can choose n such that U^Jyp)^Vf^. 
Since p ^ { S \ { p ) y ^ V for this n, we can choose m satisfying V^^W) ~  oj- In 
consequence we have On the other hand, it follows from p ^ V ^
(/3<a) and from i) that n (/5<a). This implies
and consequently p^M. ,^^n,a> i.e. ^  covers R.
Since is obvious, we can conclude, from Michael's theorem, the para-
compactness of R. Thus it follows from A. H. Stone’s theorem.®  ^ that R is fully 
normal.
To complete the proof, let us show that {S (p , © ^ )lw = l,2 , for =
{ ( 5 ^(3/))° I y e i? }  is a nbd basis of each point p  of R. Let U{p) be an arbitrary 
nbd of p , and choose n satisfying U^i{x)^U(x), then we can find m ^ n  such that 
UrAx)(^Sl{x). If (5 2 (3;))°3 a;, then considering S l{y ) r,Sl(x)=i=^, we have y^U ,,,{x) 
^ S l[x )  from i). Hence it follows from ii) that Sl^{y)^Sl(y) ^  U^ix) because we 
can assume, without loss of generality, that m > n  implies 5 ^(3/) ^ 5 (^3/) for every 
y ^ R .  Therefore we have S(;\;, ^ni)^U n{x)^U (x), i.e. {S {p , |m =l,2, • • •«} is a 
nbd basis of p. Thus we conclude the metrizability of R  b}^  the metrization 
theorem of P. Alexandroff and P. Urysohn.
Now we turn our attention to the application of this theorem. Some of the 
following theo’^ ems are well-known, and some of them are probably unknown.
T h fo r e m  2 . (Y u . Smirnov and the authory^ A regular space R is mMrizahle 
i f  and only if  there exists an open basis 58= 53^  ^ of R, where each $8^  is a locally 
finite collection of open sets.
Proof. W e  define nbds Un,nM mi )^ ^f point x of R  for every
pair n ,m  of positive integers as follows : Let V (^oc) =  n {V lx ^ V ^ V j^ } .  Then 
it x ^ U ^ U ^ V ^ (x )  for some [ / e we let U,^^ f,,(x) =  V,^(x),Sl^^^(x) =  U; otherwise 
we let U‘,,^^(x)=Sl^^(x)= R. Moreover we put
__________________ Sj  „ (X)=Vn (X) n [n[(VY\x ^ V ,  V ^  SS„}].»>
4) If S^ i^p) intersects no set of Ttm,n-
5) Full normality and paracompactness are equivalent in To-spaces. See [11].
6) S ( p , U ) = ^ { U \ p ^ U ^ U } ,  S (A , V i ) ^ ^ { U \ U  U ^ U }  for a covering IX, a 
point P and a subset A.
I) See [10] and [7].
8) From now forth we prove only the sufficiencies since the necessities are generally 
obvious.
9) We denote by or C(F) the complement set of V.
1 8 6  J u n -iti  N a g a t a
Since and 55,^  are locally finite, we have nbds of a;
satisfying the conditions i) ii) of Theorem I. For every nbd U{x) of an arbitrary 
point a; of we can choose n, m and such that
U { x ) ^ V , , [ x ) ^ V ^ V ^ x  
because R  is regular and 55 is an open basis of R. Hence U^ r^n[^ ) =  V,^{x)^U{x)y
i.e. {t/^ ^^ ;,^ (A;) l# ,w = l,2---- }  is a nbd basis of a;. Thus we conclude, by Theorem
I, the metrizability of R.
T heorem  3. [K. Moritay^"  ^ A T-^space R is metrizable i f  and only i f  there 
exists a countable collection =  1,2 — • •} oj locally finite closed coverings o f R
such that S(x, ^n)^U{x) for any nbd U{x) of any point x o f R and for  some n.
Proof. Since  ^ is locally finite, N.^{x)= is an open nbd
of any point a; of R. If S{x, we let
Un. m (X) =  N „  (X), S l ^  (x) =  (X).
If S [x, I? )^ ^  (x), then we let
U n , m {^ ) =  5 ,^ m (^ ) “
Moreover we let generally Sl^^{x) =  N,y,(x).
Then U^ -^^ {^x), Sl^^{x), Sl^^{x) are nbds of a; satisfying the conditions i) ii) 
of Theorem I. Since covers R, N^ (^y)<^S{y, for every y ^R]  hence y ^  
U„„„{x) implies S lJy )= N Jy )^ S {x , ^,n)YQ{NJx)Y = (S lJx)Y . 
y ^ S l ^ i x )  implies
S l,„  (y) =  N,„ (y) C  ix) C N„ (x) =  (a;).
Thus this theorem is a direct consequence of Theorem I.
Theorem  4. (A. H. Frinky^'  ^ A T^space R is metrizable i f  and only i f  one
can assign a nbd basis {F^ (^a;) ]^=1, 2 ---- }  for every point x oJ R such that for
every n and x ^ R  there exists m—m[n,x) satisfying the condition: V „^ {x) nXy) 
4=<5?) implies F,,,(3;)^ (a;).
Proof. Letting ^n =  [Vn [x)\x^R], =  [Vni{n,x){ ]^ ^ R ] ,  we define nbds 
Un{^),  ^ f^ a; of by
Un (X) =  S (x,  »„), S* (X) =  S (X, 2B„), SI (X) =  ,) (x).
If (v) ^ (%) 4=^, then there exists such th at a; e  (z) $(F„,(„,y) (y)Y.
In the case of m[n,z)Sm[n,y) we have ym{n,y){y)^V^[z)^x since we can assume, 
without loss of generality, that m'^ n implies F^ (^a;)^ F^ (a;). Therefore
si(y) =  v„^ n,y) (y) C  S (a;, S ,) =  l\ (x).
In  the case of m(n, z)^m(n, y) we have xe.V„,{n,,)(z)‘^ V (^y) and consequently
Sl(y) C  F„(^) C Un (X).
F or a given nbd U(x) of x  we take n with V„(x)q U(x). Then we have
10) See [5],
11) See [3].
A contribution to the theory of metrization 187
S  {X. [X) ^ U ( X ) .
Hence this theorem is also a direct consequence of Theorem I.
T h e o r e m  5 . [the authorY'^ "^  A Ti-space R is metrizable if and only if  there 
exists a family {fo^\cL^A} of real valued functions of R such that 
i) n /)3 continuous for every B ^ A ,
ii) for any nbd U{x) of any point x of R there exists oc^A and a real number 
Proof. We let^(j\;)= ■ c(.\f^ {x) < ■ for each point a; of R. For every
188 Jun-iti N a g a t a
2
pair r<r' of rational numbers we let
Uf,f' {%) = {3^1  ^ A [y) < r'j n bl n /c (y) > rj.
CH^A(X) a^C(A(x))
S'r.r' (X) =  {y\
J'l n L  iy) >=C(AM)
r + r ' r ' —r
Let p(^Ury(x) (x^S^R) and l e t /„(/>) >r' (xe:S^),  ^ /» (P) (x ^ S.,).
a^C{A(x))
Then we can choose a(X)^A(x) for every x^Sj  ^ such that /^ (^ ) [p) + ^  ^
2 ' 3
i<r'). Hence we have, by the continuity of  ^ fa{x)iP)y  ^ i^ d^ U{p) of p satisfy-
X^Si
ing
In the same way we can choose cl[x)^C [A[x]) for every x ^ S 2, and a nbd V{p) 
of P satisfying
Thus we get a nbd U[p) r^V{p) of p  with
U{P)r.V(P)r.S\Ax)=4> (^eS).
Similarly we have, by the continuities of fa  and f^, a nbd W{p) for any 
point P such that p^S\^ ,^{x) implies W[p)^Ur,r'[p).
To show that {Ur,r'{x)\ry r' are rational numbers with r<r'}  is a nbd basis 
of a;, we take, for a given nbd U[x) of a;, a e ^ , E with fJ,x)<E, fa{{U{x)Y)^S. 
Moreover we choose rational numbers r,r' with f ^{x)<r<r'<E. Then it follows 
from Gi^A{x) that
Ur.r' [X) C  { y |  ( y ) < r ' }  C =  { y \ f ,  [ y ) < r ' ]  C  U  (x),
asserting the metrizability ot R b}^  Theorem I.
T h e o r e m  6 . {A. H. Stone, K. Morita and S= Hanaiy^^ Let f  be a closed 
continuous mapping of a metric space R^  onto a topological space R 2 . Then R^  
is metrizable i f  and only i f  the boundary Bf~^{y) of the inverge image f~^[y) is
12) See [8].
13) We denote f a { A ) ^ £  the fact that/q.(a;)^£ {x ^ A ) .
14) See [12] and [6]. With respect to the proof of the necessity, see also, those 
papers.
compact fo r  every point y  of R^ .
Proof. Let U„,^=Si „ [ B then U^ ,^  is evidently an open set. 
We define nbds U„{x) and S\{x) of each point a; of J? by
Un [X)  = f{U „„), S l _ ,  [X) =  (C Unix))
For a given nbd U{x) of we can choose n satisfying Si/j,[Bf~'^{x)) ^f-'^ [U{x]) 
since f-' [^U{%)) is an open set containing f~'^ [%) and Bf-\x) is compact. Hence 
f{U,„,) =  U,,{%)<^ U[%), i.e. {r/^(A;)l^=l,2 is a nbd basis of
Next p^Un{oc) and p^ Sl{y) (y^ T  c  R^ ) imply f~^{p)  ^ =  </>
{x^S) and f~^{p) Q. Un+i,y [y^T); hence there exists an open set U {f~^(p)) con­
taining f~^(p) such that
=  (A;eS), U {f-\ p ))^ U ,,y  iy^ T ).
Thus we have, by the closedness of /, a nbd f{U(f~^(p))) of p  with
f{P U -\ p)))r.S l{x)^ < i>  {x ^ S ),f{U {f-^ {p )))^ U ,{y )  [y^TY  
hence R.^  is metrizable by Theorem I.
The following is a generalization of the theorem in our previous p a p e r a n d  
is a slight modification of Theorem I too.
T h e o r e m  7 . A Ti-space R is metrizable i f  and only i f  there exists a nbd 
basis [S^{x)\n = 1 , 2 ---- } of every point x of R satisfying
i) y^S.^{x) implies x^S^[y),
ii) fo r  every n and x^ R  one can assign nbds S'J^ x) S”{x) such that y^S^[x) 
implies S^(y) Sl^ (X)=(J) and such that y^S'J^x) implies S l(y)^x.
Proof. Put Unix) =  s„{x), Slix) =  S„(a;)  ^S"(^), S {^x) =  S' {^x), then y^U„ix) 
implies x^S„ {y) and consequently S"{x)r,S'„{y) =  4>. Therefore z^S^ix) c; Slix) 
implies i.e. S^(2)cS„(a;). Since y<$U„{x) implies 5^ (3^ )^ S^ (^ )qS' (^3/) ^
S'^ {x) =  <^ , we can conclude the validity of this theorem from Theorem I.
T eceo rem  A T^space R is metrizable i f  and only ij there exists a non­
negative function cp[x,y) of R x R  satisfying
i) <p {x, y)-= cp {y, x)
ii) d{x, A)  =  in f <^p {x, y)\ yeA j is, for every subset A o f R, a continuous 
function o f x,
iii) [S^{x)\n =  \ ,2 ---- } for S^ [^x) =  [y\q){x, y )<\ /n^ is a nbd basis o f any
point X o f R.
Proof. W elet == 5„+i(a;). Th.en p^U^[x) (x^A) and (y^B)
imply d{p, A )^ l jn  and d{p, B)^l/(w +l) and hence there exists a nbd S'^ (P) of p 
satisfying
d{S:  {p) , A)>\/ in+\) , i iSl ip) ,B)>Q.
In consequence
15) i nf  {p{y, x ] \ x e A } < l / n } .
16) Lemma 2 of [8].
17) This theorem is an extension of Corollary 6 of [8].
A contribution to the theory o f metrization 189
S2P)nS2x) =  4> (x^A), S" (P )^y  (y^B ).
Therefore R  is metrizable by Theorem 7.
C o r o l l a r y .  A Ti-space R is metrizable i f  and only i f  there exists a non­
negative function cp[x,y) of RxR satisfying
i) q> {x,  y ) = q >  {y ,  x)
ii)  d  (x, F) =  i n f  \cp{x,  _y) I j ' s F }  is,  f o r  every  c losed set  F  o f  R, a c o nt i n u o u s  
function of x,
iii) |S^ {x)\n = 1 ,2 ---- | for {x) =  [y\q> [x, y ) < \ /n ^  is a nhd basis of any
point X of R.
Proof. Since d{x, A) = d {x, A) is obvious for any subset A  of R, this pro­
position is a direct consequence of Theorem 8 .
The following two theorems are directly deduced from Theorem I and 
from Theorem 7 respectively and can be considered extensions of Chittenden's 
theorem.
T h e o r e m  9. A T-^-space R  is metrizable i f  and only i f  there exists a non­
negative function q){x,y) of RxR satisfying
i) q> {x, y ) = ( p  {y ,  X)
ii) for every E>0 and every x ^ R  one can assign 8(8, a;)>0 such that cp[x, z) 
<8{E,x) and cp[y> z)<8{E, y) imply (p(x,y)<6 and such that cp(x, z]<S{£, x) and 
g)[z, y)<8[e, z) imply q)[x,y)<e,
iij) =  1 ; 2 ---- } for S^{x) =  [y\(p{^, y )< l/ '^ ]  ^ '^^d basis of any
point X of R .
Proof. I t  is obvious.
T e[e o r e m  10. A Ti-space R is metrizable i f  and only i f  there exists a non­
negative valued function q>{x,y) of R x R satisfying
i) <p (x,  y ) = < p  [ y ,  x)
ii) for every 8>0 and x ^ R  one can assign Sl( ^ ) > 0  and ^ ) > 0  such that 
(p[x, z)<8i{E) and cp[y, y) imply q>{x, y)<8,
iii) [SrX^)\n = 1 , 2 ---- } for S,^{x) =  [y\q)[x, y ) < l /n ]  is a nbd basis of any
point X of R.
Proof. I t  is obvious.
The following is an extension of Alexandroff and Urysohn's theorem.
T h;e o r e m  11. A Ti-space R  is metrizable i f  and only i f  there exists a coun­
table collection {^^ \n  = \ , 2 ----} of open coverings such that
i) for every we can choose an open covering satisfying 
UJ for every closed subset A of R,
ii) [S {x, Un) \ n = 1 , 2 ----1 is a nbd basis of any point x of R.
Proof. Letting
5 , (a;) =  5 (a;, U.), S ' M  = S {x ,^ ^ ) ,  
for p^S^{x)  (x ^ A )  we have, by H J, a nbd W^{p) of p  satisfying
"^n(P) n ^ { ^ )  =4> [%^A). Since S'^{p) =  Wn(P) n^{P> evidently satisfies the
190  J u n - it i  N a g a t a
condition of Theorem 7, we conclude the metrizability of R.
T h e o r e m  12. [R. H. BingY^'* A regular space R is metrizable i f  and only 
i f  there exists a countable collection 2-• • •} of open collections such that
i) the sum of the closures of any sub collection of is closed,
ii) %^\n =  1 , 2 ---- , S[x, U^ ) =t=^ } is a nbd basis of each point x of R.
Proof. For every triad I, of positive integers and every point x oi R
we define nbds S,^,^,,,{x) and by
S , , =  S (a;, U,), == S [X, U J
if
a; e  C 5 C ^  S (x, U.),
and {x) =  =  R otherwise.
It  follows from the regularity of R  that {Sj^^i,^{x)\l, m, ^ =  1 ,2  • •••} is a nbd 
basis of a;.
then
P ^ S  {A, U,) 3  =  sT^Tu^),
which implies V {p) r^Sl^^^Jx)=<p (x^ A )  for some nbd V (p) o ip . If P^Sl^^^^(y) 
[y ^ B ),  then
P ^ S  [B, U..) 3  =  5^2757),
which implies W {p )^ y  [y ^ B )  for some nbd W {p) oi p . Letting S'i[^^^{p) =  V{p) 
n^{P )>  we have three nbds satisfying the condition i), ii) of Theorem 7. Thus 
this theorem is also a direct consequence of Theorem 7.
With respect to a topologically complete space^®\ we have a simpler con­
dition for metrizability than those of Theorem I and Theorem 7.
T h e o r e m  13. A topologically complete space R is completely metrizable i f  and
only i f  one can assign a nbd basis [U ,^[x)\n=\,2---- } for every point x of R such
that for every n and each point x of R there exist nbds S (^j )^, S^ j^ x) of x satisfying 
that y „{x) implies Sl(y) r,Sl[x)=(j).
Proof. If we review the proof of Theorem I, then we know that the con­
dition ii) was not used to prove the fullnormality of R. Hence R  is fully nor­
mal in the present instance. Since R is fully normal and topologically complete,
there exists, by N. A. Shanin’s theorem2«^ a countable collection {% ^\n=\,2---- }
of open coverings having the following property: If a maximum filter 
CL^Ay of closed sets is divergent, then we can choose  ^ satisfying S{x, 
for every x ^ R  and some FqjGS.
On the other hand let {(S^(A;))°|A;ei^}, then for any x, y  with x ^ y  we
18) See [I].
19) A Tg-Space R  is called a topologically complete space if R  is homeomorphic 
with a Gs -set of some compact Tg-space.
20) A Ta-space R  is topologically complete if and only if there exists a countable
collection {U^ |^  ^=  l ,2 , -----of open coverings such that if a maximum filter ^ of closed
sets is divergent, then we can choose Uyi with U^<SiS={F^ See [9].
A contribution to the theory o f metrization 191
have [x]  ^ (3;) == ^ for some m, n. Let us assume m ^ n ,  then x^[Sl^{z)Y  
implies z^U^{x)  and consequently z^Uy,{y) because we can assume without loss 
of generality, that m ^ n  implies (x ^ R ).  Hence Sl {^z) Sl(y) = ^ ,
which means In the same way we can show a;$ 5 (3^ , 0^ ).
Letting we have a countable collection =  1 , 2 -----}  of
open coverings having the property of {53^} and that of {©„}. Now we shall
show that {5 (jv, SBniASfBwaA---- ASB^ ^^ ) 1^,-=1, 2 ------{ i = l , ------ k), k = = l ,2 ----- } is a
nbd basis of any point a; of R. If we assume the contrary :
S { x , m n , )  n S { x , m n , )  n •••• n 5 (a;, 2Bn,) =  s  (a;, SB ,, a  •••' ^  ^ n j , )  ^  U ( X)  
for some open nbd U{x) of a; and every n,, k, then [{U(x))^', S{x, 2 B J| n = l,2  •• •} 
= ®  is a family of closed sets having the finite intersection property. Since all 
the sets of ® have obviously no common point, if we construct a maximum filter 
I? of closed sets such that ^  is divergent. Hence we can choose
satisfying S {x ,^ ^ )^ F ^  for some F e S .  This contradicts 5(a;, 2B ;,)g @ ^ § since 
?? is a filter. Thus we have the countable collection {SB^i A • • • • ASBw^  =  1,2* • • •
( '^=1^---- ,k ) ,k = l ,2 ----- } of open coverings such that { 5 (a;, SBwiA ----- ASB^ )^} is a
nbd basis of a;, which combining with the full normality of R induces, by Alex- 
androff and Urysohn’s theorem, the metrizability of R and consequently the 
complete metrizability of R by E. Cech's Theorem^i).
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