U NIFORM SPACES WHERE ALL C O NTINUITY I S UNIFORM

KyungpOOK M&l.hematical Joumal Vol.34, No.2, 247.'257, Oecember 1994

U NIFORM SPACES WHERE ALL C O NTINUITY I S UNIFORM

Nilson C. 8ernardes Jr. and Oinamérico P. Pombo Jr.*

1.

Introduction

In his seminaI memoir

[1 0 J ^,

^Weil has proved that to every uniformiz- able space (X

,

C) one can associate a uniform space (X

,

Uc) (Uc ^being called the universal unifornlity associated with the topology C) such that the topology induced by μ'c on X is C and such that every continuous function on (X

,

_Uc) (with values in any uniform space) is uniformly con- tilluous. Such an interesting fact shows that the following problem

,

whose discussion is the main goal of the present article

,

is natural and relevant:

Under suitable collditions on the uniform space X

,

to characterize the property that every continuous function from X into an arbitrary uniform space is uni[ormly continuous

In order to reach our purpose we introduce the class of ultranormal topological spaces

,

which properly contains that of paracompact spaces (proposition 4 and example 1) and is properly contained in that of collec tionwise normal spaces (proposition 3 and example 2). As a consequence

,

every u비l니l“tra뻐no아rmé녀때‘a혀i페1 s얘pace IS U띠U Ol“fωo아rm메mιzab비le. Our main result i잉s t“매heorem 1’

w

빼 .. h꾀lC얘h fl“urrπun뼈l

a paracompact

,

an ultranormal) uniform space

X

^to satisfy the desired property. We also construct examples which show the need for the hy- potheses used in our central theorem. We c10se the article by establishing theorem 2

,

which subsumes the fact that if each continuous function from a uniform space into

[0 ,

1 J is uniformly continuous

,

then the same holds when

[0 ,

1 J is replaced b띠ya뻐n arbi씨Itr대ary precom껴C야t s얘pace e

Throughollt we shall adopt the terr

Received Jalluary 7, 1993

* Partially supported by CN P

,

24í

248 Nilson C. Bernardes Jr. and Dinamérico P. Pombo Jr

Definition 1. Let π be a ∞vering of a set X. A refinement R' of π

,

^s

said to be subordinated to R if the following condition holds:

If 끼， V2 E R' and

ví n

V2 ￥

0 ,

then for each x E 끼 and for each y E

V;

there is a U E R such that {x

, y}

C U.

Definition 2. A topological space X is said to be quasi-ultranormal if every open covering of X has an open subordinated refinement. A topological space is said to be ultranormal if it is quasi-ultranormal and separated

Proposition 1. Every c/osed subspace 01 a quasi-ultranormal spa.ce is quasi-ultranormal

Proof Let X be a quasi-ultranormal space and let F be a c10sed subspace of X. Let R be an open covering of F and

,

for each U E π， let Vu be an open subset of X with U ⁼ F

n

VU . By assumption

,

the open covering

R = {Vu ; ^{U E R} U {X - F}}

of X has an open subordinated refìnement R'. We c1aim that

R

’

= {FnV

;V

ER'}

is an open refinement subordinated to R. lndeed

,

let U₁ = F

n

V

,

^and

U2 = F

n

V2 ( 끼

'V?

^{E R') be} intersectin~ elements of R'

,

존nd let x E U" Y E U_{2 .} Since R' is subordinated to R

,

there is a V E π such that

{x ， ν}

c

V. Hence {x

,

y} C F

n

V

,

which proves that R' is subordina.ted

t。 π and concludes the proof^‘

Proposition 2. Let X be a lopological sψace. [1 eve7.Y open subspace 01 X is qua.si-ultranonnal

,

then eve7"y subspace 01 X is quasi-ultmn07‘m띠.

Proof Let A be a subspace of X and let R be an open covering of A. For each U E π， let V

u

be an open subset of X such that U = A

n

^V

_u .

Let

R = {Vu;UE R}

,

a.nd take A as the open subset UUEπVu of X. Since R is an open covering of A

,

there is an open refìnement π， subordina.ted to R. Arguing a.s in the proof of proposition 1 we show tha.t

R' = {A

n

V

;\I

E R’},

is an open refÌnement subordinated to π， as was to be shown

Uniform spaces where all continuity is uniform 249

The following propositions guarantee that the class of ultranormal space is situated between tbe classes of paraιompact spaces and collec- tionwise norma.l spaces

Proposition 3. Every u/tmnormal space ís colleclionwíse normal (hence normal). In particular

,

everν ultmnormal space ís unífoηnizable.

Proof Let X be an ultranormal space and let (Fi)iEI be a discrete fam.ily of c10sed subsets of X‘ För each i E 1‘ put

Ui = X - Uj;é;껴

Since (F;)iEl is locally lìnite

,

eacb U‘ is open in X ( 띠， chap

.I, 31 ,

propo- sition 4). By hypothesis

,

the open covering π = {α; i E

I}

of X has an open subordinated relìnement 1(.'. For each i E 1

,

let

V;

be the union of all elements of 1(.' which meet Fi; ^끼 is a neighborhood of Fi' It remains to show that V.‘

n

끼 =

ø

if i ￥ j. In fact

,

Sllppose that W

,

E π， meets Fi" vV2 E π'meets F;

,

(where;1

'f

;2) and W1 nw

, 'f 0.

Let ψ1 E w1nFi

,

and 1.L‘2 E W

, n

^Fi

,.

Then there is a k ε 1 such that ^{W1 ψ2} C Uk . But tbis is an absurd

,

since each U_i meets at most one 건

Proposition 4. Everν pamcompact space is ultranormal. ln particular

,

every metrizable space is u/tmnormal Proof Immediate from theorem 2 of [9]

Now we shall give an example of an llltranormal space which is not paracompact. For this purpose

,

we shall consider a type of space stlldied by Dieudonné in [4] and [5]. In the seqllel we shall also see an example of a col\ectionwise normal space which is not ultranorma.l

Example l. Let X_o be an llllcountable well-ordered set witb a greatest element. Let a be the smallest element of X_o and let b be the smallest of the elements

x

E

X o

^{for which tbe interval}

[a , x[

is uncountable. Let

X

denote tbe interval

[a ,

이 endowed with the separated topology generated by the set of allleft half-open intervals

,

bounded or not. We assert that X is ullranormal‘ Indeed

,

^{let 1(. be an open covering of} X. For each x E X

,

let. Ix 二]ax^，bx] be an int.erval sllch that x E Ix and Ix is contained in some element of 1(.. By [4

],

since

U

=

UxEx(Jx X 1•)

is a neighborhood of the diagonal /:; of X x X

,

there is a y E X sllch that

250

Nilson C. Bernardes Jr. and Dinamérico P. Pombo Jr

U:::) [y ， b[x[ν ， b[. Therefore

(*) UxEKIx :::)

[y , b[ ,

where

I<

= {x E X;y E Ix}. For each x E

I<,

put J_x =]y

,

b_x]'

(J

x)xEK is an open covering of ]y

,

b[

,

J_x C Ix(x E

I<)

and

,

if Xl

,

X2 E K there is an

X3 ε

I<

with J_X3

그

JX ] U JX2 (by

(*)).

Now

,

by the compactness of [a

,

y]

[4] ,

the open covering

(Ix

n

^[a

,

ν])xEX

of [a

,

ν1 has an open subordinated refinement

n.

^{Since [a}

^,

^씨(

=J ^•,

^{ν]) is}

open in X

,

the family

n'

⁼

nU {J

x}xEK

is an open covering of X which is subordinated to

n.

^Hence

^,

^{X is ultra}

norma.l. However

,

X is not pa.ra.compact [5J

Rema1.k 1. a.) A product of ultranormal spaces is not necessa.rily ultra.nor- ma.l. For insta.nce

,

let S be the set

[ 0 ,

+∞[ endowed with the topology for which the neighborhoods of a. point x are the su bsets of S which conta.in some right까tt h a.외lf-open in뼈1

t“ln뼈l

proposition 3

’

S x S is nol ultra.normal

b) It is knoι n that the product of a. pa.racompa.ct spa.ce a.nd a compa.ct space is paracompact [5J. This result is no longer true if “pa.ra.compact" is repla.ced by “ultra.norma.l". For insta.nce

,

the product Y of the ultranorma.l space X considered in example 1 by its Alexandroff compactiβca.tion (X is clea.rly loca.lly compact) is not normal ([2

],

cha.p. IX

,

34; exercise 9b). By proposition 3

,

Y is not ultra.normal. Also

,

as in a) a^,bove

,

Y is a. product of ultranormal spaces

Definition 3. A uniform spa.ce X has the uniform continuity property with respect to a uniform spa.ce Y if every continuous ma.pping from X into Y is uniformly continuous

A uniform space has the unifOl 미 continuity property if it ha.s the uniform continuity property with respect to every uniform space.

Clea.rly

,

every discrete uniform space has the uniform continuity prop erty. Also

,

by a cla.ssica.l result

,

every compa.ct space ha.s the uniform continuity property

We now sta.te our main result

,

in whose statcment “a non-trivial uni- form space Z" is characterized by the property that its uniformity is dif- ferent from {Z x Z}

Uniform spaces where all continuity is uniform 251

Theorem 1. Let X be a uniJorm space and consider the following prop- erttes:

(i)

X has the unμorm continuity property with respect to some coπvex c/osed non-trivial uniJorm subspace of a locally convex space;

(ii) X has the uniform continuity property;

(iii) Every equicontinuous family of mappings Fom X into an arbitrary uniJorm space is uniJormly equicontinuous;

(iv) The vicinities of X are all πeighborhoods of the diagonal ^b, in the product space X

x

X ;

(v )

For each open covering oJ X lhere is a vicinity U of X such lhat

,

fOI' each X E X , U(x) is conlained in some element of lhe covering.

The following implicalions are always ^t1'Ue:

(v) =추 (iv)

각

(iii)

섭 “

^{i) =} (i).}

JJ X is quasi-u/tranormal

,

then (ii) through (i띠 are equivalenl.

Jf X is pamcompact

,

lhen (ii) through (v) are equivalenl.

Jf X is metrizable

,

then 떠 through (v) are equivalenl Proof Let U denote the set of vicinities of X

(v) =추 (iv) ^‘ Let U be a neighborhood of b, in X x X . For each x E X there is an open subset Ux ^of X such that x E Ux ^and Ux ^x Ur ^C U. Since (Ux)xEX is an open covering of X

,

the hypothesis ensures the existence of a U' E μ such lhal each U'(x) is contained in some Uy. Consequently

,

UrEX(U

’

(x) X U'(x)) C UxEx(Ux x Ux) C U

Now

,

if (y

,

z) E U

’,

then (y

,

z) E U’(ν) x U'(y). Therefore U' C U

,

which proves that U E U

(iv) =} (iii) Suppose that Y is a uniform space and that H is an equicontinuous family of mappings from X into Y. Let V be a symmetric vicinity of Y. By assumption

,

for each x E X there is an open neigh borhood Ur ^of x such tha.t

(f

(x),f(x')) E V for x' E Ux ^and f E H. Therefore

(J

(z)

,

f(z')) E \12 for z

,

z

’

E Ux ^and f E H

,

that is

,

(f

x J)(Ux x Ux) C V² for

f

E H and x E X

The set U = UxEx(Ux ^X Ur ) ^{is a neighborhood of b, in} X x X

,

hence an element of μ (by hypothesis)

,

and

(f

x J)

(U)

C

V

² for every

J

^E

H .

Thus H is uniformJy equicontinuous

(iii) =수 (ii) =수 (i) : Obvious.

(ii) =} (iii) : Suppose that X has the uniform continuity property and let

(J

;);El be an equicontinuous family of mappings from X into a uniform

252 Nilson C. Bernardes Jr. and Dinamérico P. Pombo Jr.

space

Y .

^{For each vicinity}

V

^of

Y ,

^put

Uv =

{(J,

g) E F

(I

j Y) x F

(I

j Y)j

(J

(i)

,

g(i)) E V for all i E

I} ,

where F

(J

j Y) denotes the set of all mappings from 1 into Y. It is easily seen that

,

as V runs through the set of all vicinities of Y

,

the sets U_v form a fundamental system of vicinities for a uni미liformit따t

F(υIC’ Yη) endowe려d 삐ítw비th U'. Define h : X • F

(I

^j Y) by h(x)(i)

=

fi(X)

,

x E X

,

i E 1. The equicontinuity of

(J

i).El implies the continuity of h. Dy hypothesis

,

h is uniformly continuous

,

and therefore

(J;) .El

is uni[ormly equlcontinuous

Assume that X is quasi-ultranormal and satisfies (ii). To establish

(비 it is enough to prove that the set V of all neighborhoods of ['; is a uniformity on X (indeed

,

in this case

,

the uniformizability of X implies that the topology induced by V on X is finer than that of X). ln order to do so. it suffìces to show that for each U E V there is a μ1 E V with W² C U. Fix a. U E V. Ea.ch x E X has an open neighborhood 11. such tha.t IIx X Vx

c

U. 50 V

=

UxEX(Vx ^X IIx) belongs to V and V C U. By

assumptíon、 이1e open covering (ι )xEX of X has ^a.n open subordinated refinement (W;)iEJ. Put

W = UiEl(Wi ^X κ)

W E V and

,

if (x ， ν) E W²

,

there are i" i2 E J such that (x,y) E _(Wi1

X

Iι1)

0

(κ2 x Wi2 );

hence there is a z E Wi1

n

Wi

,

such that (x

,

z) E Wi1 ^X Wi1 and (z

, y)

E Wi

,

X ~Vi，. Therefore Wi

, n

W샤

l' ø ,

and consequently there is a ψ EX such that

(x ,

y)

E

κ X ~ι. Thus W2 C V( C U)

,

as desired

Assul11e tha.t X is pa.ra.col11pa.ct a.nd rel11ember the follo、이 ng fa.ct ([7

),

cha.p.5

,

^{theorem 28): A regula.r spa.ce} Y is paracol11pa.ct if and only if

,

^fOl

every open covering R of Y

,

there is a. neighborhood V of the dia.gonal of Y x Y such tha.t the covering (V(ν))YEI' is finer tha.n R.

Now

,

if X sa.tisfies (ii)

,

then every neighborhood o[ ['; in X x X belongs to U (for X is ultra.norl11a.l by proposition 4). By wha.t we ha.ve just

l11entioned

,

X sa.tisfies (v).

Finally

,

assume that X is l11etrizable

,

and let d be a metric on X compatible with its uniforTTÚty. 5uppose that X has the uniform continuity

U niform spaces where all continuity is uniform 253

property with re5pect to a convex cíosed non-trivial uniforrn subspace C of a locally convex space

,

and that (v) is false. Then there is an open covering R of X such that

,

for each U E U

,

there is an ^X E X such that

u (x)

is not contai necl in a.ny element of π. ln particula.r

,

for ea.ch

n

E N'

,

there is an ^Xn E X sllch that B(xn; l/n) is not contained in any elernent of π Therefore there is a Yn E

X

with

o <

d(xn

,

Yn)

<

l/n

We a.ssert that (Xn)nEN' (ancl hence (Yn)nEN' ) does no1 have a. c1uster point. ln fact

,

suppose that X is sllch a point. lf W E R is such that

X E W ancl if 8

>

0 is such that B(x; 8)

c

W

,

then there is an n E N

,

n

>

2/8

,

50 that .1:n ε B(x; 8/2). Hence

B(xn; I/n) C B(x; ⁸⁾

c

W

,

a contracliction. As a consequence

,

the sequences (Xn)nEN' and

(Yn)nEN'

are locally finite. Let U5 now con51ruct

,

by incluction

,

a sequence (mn)nεN'

of positive integers and two sequences (X~)nEN' ancl (Y~)nEN' 5atisfying the following properties for all n E N':

I

mn+l

>

m n

(**) { x~ = Xmn ancl Y~ = Ym n

l

X~ (resp. Y~) is c1ifferent from 씨 (resp. xj) if j

#

n.

For t.his purpose

,

pllt 지 2."1 ~ 피 = YI ancl m

,

^{= l.} Suppose that

,

for a certain k E N'‘ we have defined

,

for ea.ch n E N'

,

11

:s

k

,

x~， 따 E X and mn E N"

,

such that (XX) ^{is valid for 1} :S n :S k and 1 :S j :S k. Then take

1nk+¹

>

mk so t.ha.t _Xmk+₁ (resp. _Ymk+l) is c1ifferent from

yj

(resp 끽) for a.1I 1 :S j :S k

,

^a.ncl put 야+1 = Xm.I<+I' 따 +1 YmHl (such a.n mk+1 exists because (Xn)nEN' a.ncl (Yn)，εN' c10 not ha.ve a c1l1ster point)

Consicler the c1is.ioint c10secl subsets A {x~; n ^E N'} a.nd B

{씨;11 E N

‘}

of X. Let V be a. 、 icinity of C

,

V 폼 C x C

,

pick a.n elemen1 (ιV) E C X C ， (lιv) rf- V

,

ancl c1efìne

J:

A U B • C by

U U

--- r

」

=

n ”

^{if z E A}

if z E B

Since

J

is cont.inllolls

,

the Tietze-Dugunclji theorem [6] guarantees the existence of a continuo야 extension

J

of

J

to

X

such that

J(X) c

^C

254 Nilson C. Bernardes Jr. and Dinamérico P. Pombo Jr.

Thus

J

is a continuous mapping from X into C which is not uniformly contmuous

,

Slnce

(J(감)

， J

(따))

=

(μ) g

^V(n

e

^{N·) and}

ⁿ

^{l많 d(강씨) = 0}

CoroJlary. 1J X is a melric space

,

theπ lhe proJ!c1"lies (i) through (띠 slaled in theorem 1 a7'e equivalent lo (v') Every open coveηπ9 oJ X admils a Lebesgue number

ProoJ Immediate from theorem 1

,

since (v) and (v’) are obviously equiv^• alent when X is a metric space.

Remark 2. [n [3J it was obtained a particular case of the a.bove corolla.ry

,

na.mely

,

the equivalence between (v’) a.nd the condition tha.t X ha.s the uniform continuity property with respect to every metric space,

N。、.y we sha.1I give the promised example of a collectionwise norma[

space which is not ultra.normal.

Examp[e 2. Let X =

[a ,

b[ be a.s in example 1. Let Y be the set

[a ,

이

endowed with the t.opology for which the points x

#

b are open a.nd for which the sels Jx, b], where

x

runs through

[a

, 이， form a. funda.menta.1 sys- tem of neighborhoods of b. The product space Z = X x Y is coLlectionwise normal

,

but the set of a.1I neighborhoods of the dia.gonal in Z x Z is not a uniformity on Z ([2], chap. [X

,

~4 ， exercise 18b)‘ By theorem 1 Z is not ultranormal

,

because

Z

endowed w’ith its universal uni[ormit.y sa.tisfìes (ii) but does not satisfy (iv)

Rema샤 3. The properties (ii) through (v) are not equivalent for every ultranormal space. For instance

,

lhe ullranormal space X of example 1

,

endowed 、、ith the uniformity given by the fLIter of neighborhoods of [;; in X x X does not satisfy (v) (by lhe fact concerning paracompact spaces staled in the proof of theorem 1), since it is not paracompact

Rernark

4.

If X is a uniformiza.ble space for 써lich the properties (ii) a.nd (iv) are equivalent., then X is necessarily quasνultranormal. lndeed, con- s

잉ide히rXe태n떼\띠do\‘”앤‘ cC (i

끼씨11띠i)). Let 1π(. be an open covering of X. The set U = UAEπ^{(A x} A)

Uniform spaces where all continuity is uniform 255

is a neighborhood of 6. By (iv)

,

there is a V E U

,

V open in X x X

,

such that V²

c

^{U. Without} loss of generality we may assume that

V = Uxε^X(ι

x

Vx )

,

where V_x is an open neighborhood of x(x E X). Let x^), X2 E X be such that κ ， n μ2 카

ø ^,

and fix a z E μI

n

Vx,. If x E VX1 and y E μ" then

(x , z)

^{E μI X V~， and}

(z ,

y) E V놔 x VX2 ; 50

(x ,

ν) E V^2• Thus there is an A E π such that

{x , y} c

A. This sho씨 s that (μ )xEX is an open refinement subordinated to π ， and therefore X is quasi-ultranormal

We now give an example of a uniform space whose topology is metriz- able for which the properties (i) and (ii) are not equiva、

Example 3. Let X be an ir페lite discrete uniform space, and let (j,‘)iEl

be the family of all mappings from X into

[0 , 1]

(each λ is necessarily continuous). Denote by μ the set of vicinities of X and by U' the initial uniformity on X with respect to the family (j

i)

‘^E[' Clearly

,

U' is compat- ible with the topology of X. Moreover

,

by construction

,

(X

,

U

’)

has the uniform continuity property with respect to

[0 , 1].

Noκ since

(X , U')

is precompact ([1], chap. 11

,

34

,

proposition 3) and since X is infinite

,

the diagonal of X x X does not belong to μ， ([1], chap. II

,

34ι theorem 3) Consequent\y

,

αis strictly coarser tban U. Thus the identity mapping from (X μ') into (X

,

U) is continuous but is not uniformly continuous

,

and so (Xμ') does not satisfy (ii)

Note that

,

since (X， μ') is topologically discrete

,

tben it is locally com- pact

,

paracompact‘ locally connected and topologically metrizable. Fur thermore‘ (X

,

μ') is also σ compact if X is countable. The above example shows that

,

probably

,

the bypothesis of metrizability on X (under which

,

by theorem 1

,

(i) tbro떼1 (v) are equiva.lent) cannot be considerably weak- ened

Theorem 2. Lel X be a uniform space. [f X has the uniform continuity property with

,

-especl to some convex closed non-trivial un ψrm subspace C of a locally convex space, then χ has the uniform contin uity propertν

u’ith

,

-especl to every precompact space

Proo

f.

Let us begin by proving that X has the uniform continuity property with respect to every compact metric space. For this purpose

,

let f{ be a compacl rnetric space. Let (ι )iEl be the family of all continuous mappings from f{ into C and let μ be the initial uniformity on κ 、、까 h respect to the family (gi)iEf. \까 c1aim that μ is compatible witb the topology of f{,

256 Nilson C. Bernardes Jr. and Dinamérico P. Pombo Jr.

which implies that U coincides with the uniformity of I<. lndeed

,

since the topology induced by U on f{ is coarser than the one of I<, let us prove the converse. In order to do so

,

let V be an open neighborhood of a point ko E f{. Let W be a vicinity of C

,

W ￥ C x C

,

and fix an element (a^,b) E C x C, (a^,b) tj W. The Tietze.Dugundji theorem asserts that there is an i E 1 such that 9‘(ko)

=

^{a and gi(k)}

=

b if k E I< - V. Thus g;'(W(a)) is a neighborhood of ko with respect to the topology induced by μ on I<, and 파 '(W(a)) C V. Now, define 9 : I< • C^l by

g( k) = (ι (k) )쉰 (k E f{),

and consider C^l endowed with the product uniformity. Tf k" k₂ E I<, k

,

^{￥ k}2

,

there is an i E 1 such that gi( 서 )

-1

gi(k_{2 )}

,

by the Tietze.Dugundji theorem^‘ consequently 9 is injective. So

,

by whal we have seen above

,

it follows that 9 is an isomorphism from f{ onto the uniform suhspace g(^f{) of C¹. Under this identification

,

every mapping

f :

X • I< can be written as

f(x) = (ι (x) )iε1 (x E X)

,

where each fi is a ma.pping from X into C. In addition

,

if f is continuous

,

then each fi is continuous

,

hence uniformly continuous (by hypothesis);

therefore

f

is uniformly continuous. Thus X has the uniform continuity property with respect to I<.

Now

,

let P be a precompact space and P the Hausdorff completion of P. As a particular case of w hat we have just proved

,

^X has the uniform continuity property with respect to

[0 ,

1]. Hence we can apply the Tietze Urysohn theorem and argue as above to show that P (and therefore P) ca.n be identified with a. uniform subspa.ce o[

[0 ,

1

F.

As a consequence

,

e、'ery continuous ma.pping from X into P is uniformly continuous

,

^a.s was to be shown

Reηwrk 5. Exa.mple 3 furnishes a uniform space X which satisfies the hypothesis of theorem 2 (X has the uniform continuity property with respect to

[0 ,

1])

,

which does not have the uniform continuity property with respect to a certain loca.lly compact Q'.compact uniform space whose topology is metriza.ble

References

[1] Bourbaki, N., General Topology, Part 1, Springer. Verlag (1989)

U niform spaces where all continuity is uniform 257

[2J Bourbaki, N., General Topology, Part 2, 5pringer-Verlag (1989)

[3J Chaves, M. A., Spaces where all conti

’

^{I.uil.y is uniJorm, Amer. Math. Monthly}

92(1985),487-489

[4J Oieudonné J., Un exemple d’espace

’‘

^{ormal n071} suscetiblc d 'une stπ ctuπ unifonne d’espace complet, C. R. Acad. 5ci. Paris 209 (1939), 145-147

[5J Oieudonné J., Une gén.éralisation. des espaces compacts, J. Math. Pures App!

23( 1944), 65•76

[6J Ougundji, J., An exiens

‘

^{on 0/ Tielze’'s iheorem, Pacific J. Math. 1(1951),353-367}

[끼 Kel!ey‘ J. L., General Topology, Van Nostrand R.einhoJd Company (1955) [8J 5orgenfrey, R. H., 01/. ihe iopological product oJ paracom.paci spaces, Bul!. Amer

Math. 50c. 53(1947), 631-632

[9J 5tone, A. H., Paracompacintss and produci spaces, BuJ!. Amcr. Math. 50c 54(1948),977-982

[10J WeiJ, A., Sur les Espaces à Siruciure UniJorme ei sur la Topologie Générale, Actual. 5cient. et lnd. 551, Hermann(1937)

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