Higher Separation Axioms, Paracompactness And

Home , Convergence space

Scientiae Mathematicae Vol.2, No. 3(1999), 321{335 321

HIGHER SEPARATION AXIOMS, PARACOMPACTNESS AND

DIMENSION FOR SEMIUNIFORM CONVERGENCE SPACES

GERHARD PREU

DedicatedtoProfessor Lamar Bentley on his 60th birthday

Received July 23, 1999

Abstract. Higher separation axioms, paracompactness and dimension are de ned for

semiuniform convergence spaces. Under the assumption that subspaces are formed in

the construct SUConv of semiuniform convergence spaces one obtains the following

0 0

results: 1 Subspaces of normal (symmetric) top ological spaces are normal, 2 sub-

spaces of paracompact top ological spaces are paracompact, and 3 for some dimension

functions, the dimension of a subspace is less than or equal to the dimension of the

original space, where these dimension functions coincide with the (Leb esgue) covering

dimension for paracompact top ological spaces. Additionally, Urysohn's Lemma, Tietze's

extension theorem and some other extension theorems are studied not only in the realm

of top ological spaces. Last but not least the interrelations b etween nearness spaces and

semiuniform convergence spaces b ecome apparent.

0. Intro duction

It is well{known that the construct Top of top ological spaces has several de ciencies,

e.g. there do not exist natural function spaces in general (i.e. Top is not cartesian closed)

and quotients are not hereditary (cf. [7; Thm. 2]). Furthermore, the formation of subspaces

in Top is not satisfactory as the following example shows: Though a p oint or the closed

pointinterval [0; 1] are distinguishable (they are non{homeomorphic), the top ological spaces

obtained from the real line IR by removing a p ointor[0; 1] are not distinguishable (they are

homeomorphic). Additionally, uniform concepts such as uniform continuity, completeness

or uniform convergence cannot b e explained in the framework of top ological spaces. In

ConvenientTop ology (cf. [16]) all these de ciencies are remidied byintro ducing semiuni-

form convergence spaces (cf. [15]). In particular, one obtains non{isomorphic spaces in the

ab ove example provided that subspaces are formed in the construct SUConv of semiuni-

form convergence spaces instead of Top. This b etter b ehaviour of subspaces in SUConv

leads to much nicer results even for classical top ological concepts such as paracompactness,

normalityorcovering dimension as will b e explained in the following:

According to Katetov [10] lter spaces can b e describ ed in the framework of merotopic

spaces. On the other hand, lter spaces have also b een describ ed in the realm of semiuniform

1991 Mathematics Subject Classi cation. 54A05, 54A20, 54E05, 54E15, 54E17, 54B05, 54D15, 54D18,

54F45, 54F60, 18A40.

Key words and phrases. Complete regularity, normality, full normality, paracompactness, dimension,

semiuniform convergence spaces, lter spaces, convergence spaces, top ological spaces, merotopic spaces,

nearness spaces, uniform spaces, maps into spheres, bire ective sub categories.

322 GERHARD PREU

convergence spaces (cf. [15]). Thus, they form the link b etween semiuniform convergence

spaces and merotopic spaces. From this fact pro t the de nitions of complete regularity,

normality, full normality (resp. paracompactness) and dimension for semiuniform conver-

gence spaces whichareintro duced in this article (complete regularity of lter spaces has

b een studied earlier in [2] and [3]). It turns out that all higher separation axioms stud-

ied here are hereditary and that for the small and the large lter{dimension intro duced

here, the dimension of a subspace is less than or equal to the dimension of the original

space, where these dimension functions coincide with the (Leb esgue) covering dimension

for paracompact top ological spaces.

Additionally, Urysohn's Lemma and Tietze's extension theorem are stated for normal

lter spaces, and an extension theorem for Cauchy continuous (resp. uniformly continous)

maps into spheres is related to the small lter dimension (resp. small uniform dimension)

of normal lter spaces (resp. uniform spaces).

Nearness spaces intro duced by Herrlich [6] form an imp ortant sp ecial case of merotopic

spaces and several results on them are needed for the ab ovementioned theorems on semi-

uniform convergence spaces.

The terminilogy of this article corresp onds to [1] and [13].

1. Preliminaries

For the convenience of the reader some basic de nitions are rep eated.

A semiuniform convergencespace is a pair (X; J ), where X is a set and J a set of lters

X X

0 0

on X X such that 1 x_ x_ 2J for each x 2 X (wherex _ = fA X : x 2 Ag), 2

0 1 1

G2J whenever F2J and FG,and3 F2J implies F = fF : F 2Fg2J

X X X X

(where F = f(y; x): (x; y ) 2 F g). A map f :(X; J ) ! (Y; J )between semiuniform

X Y

convergence spaces is called uniformly continuous provided that (f f )(F ) 2J for each

F2J . The construct of semiuniform convergence spaces is denoted by SUConv.

A uniform space (X; V ) in the sense of Weil [21] may b e considered to b e a semiuniform

convergence space (X; J )withJ =[V ]where[V ]=fF 2 F (X X ): FVgand

X X

F (X X ) denotes the set of all lters on X X . In this case (X; J )isalsocalleda

principal uniform limit space. A symmetric top ological space (= R {space) (X; X )maybe

considered to b e a semiuniform convergence space (X; J ), where J = fF 2 F (X X ):

X X

there is some x 2 X with FU (x) U (x)g and U (x) denotes the neighb orho o d lter

X X X

of x 2 X w.r.t. X . In this case (X; J ) is called a topological semiuniform convergence

space.

Every semiuniform convergence space (X; J ) has an underlying lter space (X; ),

X J

where = fF 2 F (X ): FF 2 J g and F (X ) denotes the set of all lters on X

J X

(rememb er that a lter space is a pair (X; ), where X is a set and F (X ) contains

all lters x_ with x 2 X and for each F 2 , all lters G on X with F G;amap

0 0

f :(X; ) ! (X ; )between lter spaces is called Cauchy continuous provided that

f (F ) 2 for each F2 ). Furthermore, a lterspace (X; )may b e considered to b e a

semiuniform convergence space (X; J )whereJ = fF 2 F (X X ) : there is some G2

with GG Fg. In particular, a semiuniform convergence (X; J ) is called Fil{determined

provided that J = J .

A symmetric Kent convergencespace isapair(X;q )whereX is a set and q F (X ) X

0 0

suchthat1 (_x;x) 2 q for each x 2 X ,2 (G ;x) 2 q whenever (F ;x) 2 q and GF,

0 0

3 (F\x;_ x) 2 q whenever (F ;x) 2 q ,and4 (F ;x) 2 q and y 2 fF : F 2Fg imply

HIGHER SEPARATION AXIOMS 323

0 0

(F ;y) 2 q (symmetry condition). A map f :(X; q ) ! (X ;q )between (symmetric) Kent

convergence spaces is called continuous provided that (f (F );f(x)) 2 q for each(F ;x) 2 q .

The construct of symmetric Kent convergence spaces is denoted by KConv .Every semi-

uniform convergence space(X; J ) has an underlying symmetric Kent convergencespace

.Furthermore, a symmetric Kentcon- i F\x_ 2 ) de ned by(F ;x) 2 q (X; q

X J J

X X

) vergence space (X; q )may b e considered to b e a semiuniform convergence space (X; J

where = fF 2 F (X ): there is some x 2 X with (F ;x) 2 q g.

For every symmetric Kentconvergence space (X; q ) a closure op erator cl is de ned by

cl A = fx 2 X : there is some F2F (X ) with (F ;x) 2 q and A 2Fgfor each subset A of X ;

o ccasionally, one writes A instead of cl A. In particular, X = fO X : cl (X nO )= X nO g

q q q

is a top ology on X .For each top ological space (X; X ), de ne q F (X ) X by(F ;x) 2 q

X X

i FU (x). A symmetric Kent convergence space (X; q ) is called topological provided

that q = q .

A subset A of a semiuniform convergence space (X; J ) is called closed (resp. dense)

provided that A = cl A (resp. cl A = X ). A semiuniform convergence space (X; J )

q q X

J J

X X

)

is called regular provided that for each F2J the sub lter F generated by the lter

F : F 2Fgb elongs to J , where F denotes the closure of F in the underlying base f

Kent convergence space of the pro duct space (X; J ) (X; J ) (note that SUConv is

X X

a top ological construct!). The underlying symmetric Kentconvergence space (X; q )of

a regular semiuniform convergence space (X; J ) is regular, i.e. for each(F ;x) 2 q ,

( F ;x) 2 q where F is generated by fF : F 2Fgand F = cl F ; furthermore,

X J

a symmetric top ological space (X; X ) is regular in the usual sense i it is regular as a

semiuniform convergence space. Using [5; 3.7.13. and 4.12.6.] one obtains the following

Theorem. Let (X; J ) be a principal uniform limit space (resp. a topological semiuniform

convergencespace). Then each uniformly continuous map f :(A; J ) ! (Y; J ) from

A Y

a dense subspace (A; J ) of (X; J ) to a complete, regular and separated semiuniform

A X

f :(X; J ) ! convergencespace (Y; J ) has a unique uniformly continuous extension

X Y

(Y; J ).

(Note: A semiuniform convergence space (X; J ) is called complete provided that for each

F 2 there is some x 2 X suchthat(F ;x) 2 q ; and it is called separated (or

X J

a T {space) provided that for each F 2 F (X ) there is at most one x 2 X suchthat

(F ;x) 2 q :)

By the way, a semiuniform convergence space (X; J )iscalledaT {space provided that

X 1

for each pair (x; y ) 2 X X ,(_x; y ) 2 q implies x = y .Obviously,every regular T {space

(= T {space)isaT {space (cf. [5; 3.7.12.]).

3 2

Concerning nearness spaces and generalizations the reader is referred to [13], where in the

following wesay merotopic space instead of seminearness space; in particular, Mer, Near,

Fil, Top, Top and Unif stand for the constructs of merotopic spaces, nearness spaces,

lter spaces, top ological spaces, R {spaces and uniform spaces resp ectively.

2. Complete regularity

2.1 De nitions. 1) A lter space (X; ) is called completely regular provided that for

each F 2 the sub lter G = fG X : F is completely within G for some F 2Fg

)

A subset G of a lter F on a set X is called a sub lter provided that it is a lter.

324 GERHARD PREU

b elongs to ; here F is completely within G provided that there is a Cauchycontinuous

map f :(X; ) ! ([0; 1]; ) suchthatf [F ] f0g and f [X nG] f1g, where denotes

t t

the set of all convergent lters on the closed unit interval [0; 1] endowed with the usual

top ology.

2) A semiuniform convergence space (X; J ) is called completely regular provided that

it is regular and its underlying lter space (X; ) is completely regular.

2.2 Remarks. 1) Each lter space (X; ) has a corresp onding ( lter{)merotopic space

(X; ), where = fA P (X ): for each F2 there is some A 2A with A 2Fg (cf. [17;

2.4.]), i.e. there is an alternative description of lter spaces in the realm of merotopic spaces.

2) A lter space (X; ) is completely regular i (X; ) is a completely regular nearness

space (cf. [2] and [3]).

3) According to [3] (resp. [2]) a lter spaceiscompletely regular i it is a subspace (in

Fil) of some completely regular topological space (regarded as a lter space) [note that there

is no di erence in forming subspaces of symmetric top ological spaces in Mer, Fil or Near

resp ectively]. Consequently, each completely regular lter space is subtop ological; further-

more, it is a Cauchy space [note that every comp etely regular top ological space is weakly

Hausdor (cf. [5; 3.7.12]) and that is is a Cauchy space (cf. [14; 2.13]); furthermore Chy is

bire ectivein Fil (cf. [14; 3.1])].

2.3 Prop osition. The underlying Kent convergencespaceofacompletely regular semi-

uniform convergencespaceisacompletely regular topological space (in the usual sense).

Pro of. Let (X; J ) b e a completely regular semiuniform convergence space. Then

(X; ) is a completely regular lter space, i.e. there is a completely regular top ological

space (Y; Y )such that (X; ) is a subspace (in Fil)of(Y; ), where consists of

J q q

X Y Y

all convergent lters in (Y; Y ). Since initial structures in Fil induce initial structures in

KConv ,(X; q ) is a subspace (in KConv )of(Y; q )[(F ;y) 2 q i F converges to y

S S Y Y

in (Y; Y )]. Since (Y; q ) is a completely regular top ological space and Top is closed under

Y S

formation of subspaces in KConv ,(X; q ) is a completely regular top ological space.

2.4 Corollary. A symmetric Kent convergencespace (X; q ) is completely regular (as a

semiuniform convergencespace) i it is a competely regular topological space.

Pro of. \=)". If (X; J ) is completely regular, then, by2.3.,(X; q )=(X; q )isa

q J

completely regular top ological space.

\(=". If (X; q ) is a comp etely regular top ological space, then (X; J ) is a regular semi-

uniform convergence space (cf. [15; 5.6.a)]). Furthermore, (X; )= (X; ) is completely

J q

regular by 2.2.3).

2.5 Prop osition. Every uniform spaceiscompletely regular.

Pro of. By [15; 5.9.], every uniform space is regular. Furthermore, if (X; V )isa uni-

form space and (X ; V ) a complete uniform space containing (X; V ) as a dense subspace

(cf. e.g. [19; 3.4., Satz 1]), then the underlying lter space (= Cauchy space) (X; )of V

HIGHER SEPARATION AXIOMS 325

)of (X; V ) is a subspace in Fil of the underlying lter space (= Cauchy space) (X ;

(X ; V ). Since (X ; V ) is complete, (X ; ) is the corresp onding lter space of the

underlying completely regular top ological space of (X ; V ). Thus, (X; ) is a completely

regular lter space (cf. 2.2.3)). Therefore, everything is proved.

2.6 Prop osition. Every completely regular lter spaceisregular (as a semiuniform con-

vergencespace).

Pro of. Let (X; ) b e a completely regular lter space. Since (X; ) is a subspace (in Fil)

of a completely regular top ological space, which (considered to b e a semiuniform conver-

gence space) is a regular semiuniform convergence space, (X; ) (regarded as a semiuniform

convergence space) is also regular b ecause the regular semiuniform convergence spaces form

a bire ective sub construct of SUConv and subspaces in Fil are formed as in SUConv.

2.7 Remark. The two preceding prop ositions demonstrate that for imp ortant examples

of semiuniform convergence spaces whose underlying lter spaces are completely regular the

regularity is automatically ful lled. But this is not always the case as the following example

shows: LetX b e the set IR of real numb ers and X the usual top ology on IR . De ne A

by(x; x) 2 A i 0

to X .

2 2

Put J = fF 2 F (IR ):F(fAg)g[fF 2 F (IR ): FU (x) U (x) for some x 2 IR g.

X X X

Then (X; J ) is a semiuniform convergence space such that the underlying lter space

(X; ) is completely regular (obviously, = fF 2 F (IR ): F converges in (X; X )g),

J J

X X

but (X; J ) is not regular since (fAg) b elongs to J whereas the sub lter (fAg)= (fAg)

X X

do es not b elong to J .

2.8 Prop osition. The construct CReg of completely regular semiuniform convergence

spaces (and uniformly continuous maps) is a bire ective subconstruct of SUConv and thus

atopological construct.

Pro of. Let (f :(X; J ) ! (X ; J )) b e an initial source in SUConv suchthat

i X i X i2I

all (X ; J ) are completely regular. Then (X; J ) is completely regular:

i X X

1) By [15; 5.3], (X; J ) is regular.

2) By [15; 3.10], is the initial Fil{structure w.r.t. (f ), and by [3; 3.2], (X; )isa

J i J

X X

completely regular lter space.

2.9 Remark. By 2.8., every subspace (in SUConv) of a completely regular top ological

space (regarded as a semiuniform convergence space) is comp etely regular. It follows from

2.5. that the inverse is not true, namely the uniform space IR of real numb ers is completely

regular but it is not a subspace (in SUConv) of a comp etely regular top ological space

[otherwise it would b e Fil{determined and thus it might b e considered to b e the top ological

space IR of real numb ers, which is imp ossible since the uniform structure of IR is not

t u

indiscrete (cf. [15; 5.10.])].

3. Normality

326 GERHARD PREU

3.1 De nitions. 1) A merotopic space (= seminearness space) (X; ) is called

a) regular provided that the following is satis ed:

(R) For each U2, there is some (re nement) V2 suchthatforeach V 2V, there exists

some U 2U with fX nV; U g2 ,

b) normal provided that (X; ) and (X; ) are regular where = fU 2 : there is some

c c

nite V2 with VUg

2) Let (X; ) b e a lter space and (X; ) its corresp onding ( lter{)merotopic space.

Then (X; )iscalledmerotopical ly normal (shortly: m{normal)provided that (X; )is

normal.

3) A semiuniform convergence space (X; J ) is called normal provided that it is regular

and the underlying lter space (X; )is m{normal.

3.2 Remark. Every regular merotopic space is a nearness space (cf. e.g. [13; 6.2.7. 1 ]).

3.3 Prop osition. Every normal semiuniform convergencespaceiscompletely regular.

Pro of. Let (X; J ) b e a normal semiuniform convergence space. Then (X; J )is

X X

regular. Furthermore, (X; ) is a nearness space which is normal and thus completely

regular (cf. [2]). Consequently,by 2.2.2), (X; ) is completely regular.

3.4 Prop osition. Let X be a lter space. Then the fol lowing areequivalent:

(1) X is normal (as a semiuniform convergencespace),

(2) X is m{normal.

Pro of. (1) =) (2). This follows immediately from the de nitions.

(2) =) (1). Since X coincides with its underlying lter space whenever X is considered

to b e a semiuniform convergence space, it suces to prove that X is regular. Since X is

normal as a merotopic space, it is also a completely regular nearness space. Consequently,by

2.2.2), the lter space X is completely regular. Thus, by2.6., X is regular as a semiuniform

convergence space.

3.5 Prop osition. A symmetric Kent convergencespace is normal (as a semiuniform

convergencespace) i it is a normal topological space in the usual sense.

Pro of. 1) Let (X; X ) b e a symmetric top ological space, (X; ) its corresp onding lter

space (i.e. is the set of all convergent lters in (X ; X )) and (X; ) its corresp onding

q X

nearness space, i.e. = fU P (X ): there is some op en cover O of (X; X )withOUg.

Then = :

.Thus, eachconvergent lter in (X; X )contains some U 2U. Hence, a) Let U2

fU : U 2Ug U and is an op en cover of X ; namely if x 2 X , the neighb orho o d lter

U (x)of x in (X; X )converges to x and thus there is some U 2U such that U 2U (x),

X X

i.e. x 2 U U . Therefore, U2 .

b) Let U2 , i.e. there is an op en cover O of (X; X ) with OU.IfF is a lter on X

HIGHER SEPARATION AXIOMS 327

X such that there is some x 2 X with FU (x), then there are some O 2O and some

U 2U suchthatx 2 O U .Thus, U 2U (x) F. Consequently, U2 .

2) a) Let (X; q ) b e a symmetric Kentconvergence space suchthat(X; J ) is normal.

) is completely regular and by 2.3., (X; q ) is a (completely regular) top ological Then (X; J

)=(X; )is m{normal, i.e. (X; ) is normal space, i.e. q = q . By assumption, (X;

q X X J

q q

(cf. 1)). By [13; 7.22.], (X; X )(resp.(X; q )) is a normal top ological space in the usual sense.

b) Let (X; q ) b e a normal symmetric top ological space in the usual sense, i.e. X is a

normal symmetric top ology on X . In order to prove that (X; J ) is normal it suces

) is normal since (X; ) is a lter space and 3.4. is valid. By 1), to show that (X;

(X; )= (X; ). Since (X; X ) is a normal symmetric top ological space, it is regular

X q

q q

and by [8; 4.3.5], (X; ) is regular. Furthermore, (X; ( ) ) is regular by [13; 7.2.2.].

X X c

q q

Thus, (X; ) is normal and the pro of is nished.

3.6 Remarks. 1) The underlying Kent convergencespace of a normal semiuniform con-

vergencespaceneed not benormalas the following example shows (whereas it is always a

completely regular top ological space by 3.3. and 2.3.): Let (X; X ) b e a completely regular

Hausdor space, which is not normal (e.g. the Niemytzki plane), and (X; ) the corre-

sp onding ne uniform space describ ed by means of uniform covers, i.e. is the set of

all covers of X which are re ned by some normally op en cover of (X; X ). Further, con-

sider (X; ( ) ) and let W denote the Weil uniformity corresp onding to ( ) .Then

F c F c

( )

F c

(X; W ) is a totally b ounded uniform space which is separated since its underlying

( )

F c

top ological space is the Hausdor space (X; X ). Consequently,(X; [W ]) is a normal

( )

F c

semiuniform convergence space (cf. 4.11. and 4.2.), whose underlying Kentconvergence

space (X; q ) is not normal.

2) A uniform spaceneed not be normal, namely in the following an example of a complete

separated uniform space is given which is not normal: Let IR b e the normal uniform space

IR IR

of real numb ers and IR the uniform pro duct space. Let V b e the uniformityof IR and

u u

IR IR

. put J IR =[V ], i.e. (IR ; J IR ) is the principal uniform limit space corresp onding to IR

IR IR

Then the set of its Cauchy lters is given by

IR IR

g, () = fF 2 F (IR ): F converges in IR

t IR

where IR denotes the normal top ological space of real numb ers and the top ological

IR IR

pro duct space IR is the underlying Kentconvergence space of (IR ; J IR ). Further,

= , ()

V J

(cf. part 1) of where X denotes the top ology induced by V , i.e. the top ology of IR

is not normal in the usual top ological sense (cf. [22; 21 C.5.], the pro of of 3.5.). Since IR

( IR ; ) cannot b e normal (cf. [13; 7.2.2.]). By (), this implies that the complete

separated uniform space IR is not normal as a semiuniform convergence space.

3.7 Prop osition. Every subspace (in SUConv) of a normal semiuniform convergence

space is normal.

Pro of. Let (X; J ) b e a normal semiuniform convergence space and U X .IfJ de-

X u

notes the initial SUConv{structure on U w.r.t. the inclusion map i : U ! X ,then(U; J )

is regular (cf. [15; 5.3.]). By [15; 3.10], (U; ) is a subspace (in Fil)of(X; ). Since

J J

u X

subspaces in Fil{Mer ( ) is a subspace of (X; Fil) are formed as in Mer,(U ).

X J u

328 GERHARD PREU

Then (U; ) is a normal nearness space b ecause (X; ) is a normal nearness space

J J

(cf. [13; 6.2.7. 5 and 7.2.10. 1 ]).

3.8 Remarks. 1) As is well{known subspaces in Top (resp. Top ) of normal symmetric

top ological spaces need not b e normal in general. But if subspaces are formed in SUConv

one obtains from 3.7. the following result: Subspaces of normal symmetric topological spaces

are normal.

2) The class of all normal semiuniform convergence spaces do es not coincide with the

class of all subspaces (in SUConv) of normal symmetric top ological spaces as the following

example shows: Mo dify the example under 2.7. by de ning A as follows:

(x; x) 2 A i 0 x 1.

Then (X; J ) is a normal semiuniform convergence space which isnotasubspaceofa

normal top ological semiuniform convergence space since it is not Fil{determined.

3) Since pro ducts of symmetric top ological spaces are formed in Top as in SUConv

and in Top normality is not nitely pro ductive, it follows from 3.5. that in SUConv

normality is not nitely productive.

3.9 De nition. Let (X; ) b e a lter space. Then AP(X ) is called near provided

that there is some F2 such that for each A 2A, X nA 62 F .

3.10 Theorem (Urysohn's Lemma). Let (X; ) be a normal lter space. Whenever

fA; B gP(X ) is not near, there is a Cauchy continuous map f :(X; ) ! ([0; 1]; )

such that f [A] f0g and f [B ] f1g where denotes the set of al l convergent lters

w.r.t. the usual topology on the closed unit interval [0; 1].

Pro of. Cf. [13; 7.2.6] (resp. [18]) and note:

1. fA; B gP(X )isnotnearin(X; )i fA; B g is not near in (X; ), i.e. fX nA; X nB g2

2. f :(X; ) ! ([0; 1]; )isCauchycontinuous i f :(X; ) ! ([0; 1]; ) is uniformly

continuous ( coincides with the ne uniform structure on [0; 1]).

3.11 Theorem (Tietze, Urysohn). Let (X; ) be a normal lter space, and (A; ) a

subspace (in Fil) of (X; ). Then every Cauchy continuous map f :(A; ) ! ([0; 1]; )

A t

has a Cauchy continuous extension F :(X; ) ! ([0; 1]; ).

Pro of. Cf. [13; 7.2.13] (resp. [18]) and note that subspaces of lter spaces are formed in

Fil ( Fil{Mer)asinMer.

3.12 Remark. For normal R {spaces, the classical version of Urysohn's Lemma follows

from 3.10 (note: A = A, B = B and A \ B = imply fA; B g is not near) and that

one of Tietze's extension theorem is an immediate consequence of 3.11 (note: every closed

subspace of an R {space is a subspace in Fil).

4. Full normality and paracompactness

4.1 De nitions. 1) A merotopic space (X; ) is called uniform provided that each A2

HIGHER SEPARATION AXIOMS 329

is star{re ned bysomeB2.

2) A lter space (X; ) is called merotopical ly uniform (shortly: m{uniform)provided

that (X; ) is uniform.

3) A semiuniform convergence space (X; J )iscalledful ly normal providedthatitis

)is m{uniform. regular and its underlying lter space (X;

4) A semiuniform convergence space (X; J ) is called paracompact provided that it is T

X 1

and fully normal.

4.2 Prop osition. Every ful ly normal semiuniform convergencespace is normal.

Pro of. Let (X; J ) 2jSUConv j b e fully normal. It suces to prove that (X; )is

X J

m{normal. But this is obvious, since every uniform space (as a merotopic space) is normal

(cf. e.g. [13; 6.2.7. 2 a) and 3.1.3.8. 3 ]).

4.3 Corollary. Every paracompact semiuniform convergencespaceisT , i.e. normal and

T .

4.4 Prop osition. A lter space is ful ly normal (as a semiuniform convergencespace) i

it is m{uniform.

Pro of. \=)". If (X; ) 2jFil j suchthat(X; J ) is fully normal, then (X; )= (X; )

is m{uniform.

\(=". If (X; ) 2jFil j is m{uniform, then (X; ) is uniform and thus it is normal. By

3.4., (X; J ) is normal and consequently it is regular. Furthermore, (X; )=(X; )is

m{uniform by assumption.

4.5 Prop osition. A symmetric Kent convergencespace is ful ly normal (as a semiuni-

form convergencespace) i it is a ful ly normal topological space in the usual sense.

Pro of. \=)". Let (X; q ) b e a symmetric Kent convergence space suchthat(X; J )

is fully normal. Then (X; J ) is completely regular and by 2.3., (X; q ) is a (completely

regular) top ological space, i.e. q = q . By assumption, (X; )= (X; )ism{uniform,

X J q

i.e. (X; ) is uniform (cf. part 1) of the pro of of 3.5.). By [13; 3.1.2.7.], (X; X )(resp.(X; q )))

X q

is fully normal.

\(=". Let (X; q ) b e a fully normal R {space in the usual sense, i.e. X is a fully normal

0 q

symmetric top ology on X . In order to provethat(X; J ) is fully normal it suces to show

that (X; ) is uniform since (X; ) is a lter space and 4.4. is valid. Weknow already

that (X; )=(X; ). Since (X; X ) is a fully normal R {space, (X; ) is uniform.

X q 0 X

q q q

4.6 Corollary. A symmetric topological spaceisparacompact (as a semiuniform conver-

gencespace) i it is paracompact in the usual sense.

4.7 De nition. A lter space (X; )iscalledtopological provided that it is weakly

subtop ological (i.e. (X; q ) is top ological) and complete.

330 GERHARD PREU

4.8 Remark. The construct TopFil of top ological lter spaces (and Cauchycontinuous

maps) is (concretely) isomorphic to Top .

4.9 Prop osition. A symmetric topological space is ful ly normal i it is topological and

m{uniform (as a lter space).

Pro of. \=)". Let (X; X ) b e a fully normal R {space and (X; ) the corresp onding

0 q

lter space. Obviously,(X; ) is complete; and since q = q ,(X; ) is subtop ologi-

X q

X q X

cal, i.e. (X; ) is top ological. Furthermore, (X; )= (X; ) is uniform, since (X; X )

X q

is fully normal. Thus, (X; )is m{uniform.

) is (top ological and) m{uniform. \(=". Let (X; X )beanR {space such that (X;

0 q

By 4.4., (X; J ) is fully normal, and by 4.5., (X; q ) (resp. (X; X )) is fully normal.

4.10 Corollary. A topological T {spaceisparacompacti itistopological and m{uniform

(as a lter space).

4.11 Prop osition. Every proximity space (= total ly bounded uniform space) is ful ly

normal.

Pro of. Let (X; J )beaproximity space, i.e. J =[V ] where V is a totally b ounded

X X

uniformityon X . Since (X; J ) is uniform, it is regular (cf. [15; 5.9.]). It remains to

showthat(X; )is m{uniform. Let (X; ) b e the merotopic space corresp onding to V ,

J V

i.e. = fA P (X ): there is some V 2V with A Ag, where A = fV (x): x 2 X g.

V V V

If denotes the set of all Cauchy lters in (X; ) (resp. (X; V )), then

(1) = .

V X

Since (X; ) is a contigual (= totally b ounded = precompact) merotopic space, it is

= (cf. part c) in the pro of of [17; 2.4]). ltermerotopic (cf. [10; 2.10]) and thus

Consequently,by (1), one obtains:

(2) = .

Since (X; ) is uniform, it follows from (2) that (X; )is m{uniform.

V J

4.12 Corollary. Every separatedproximity spaceisparacompact.

4.13 Prop osition. Acomplete uniform space is ful ly normal (as a semiuniform conver-

gencespace) i its underlying topological space is ful ly normal.

Pro of. \(=": Let (X; V ) b e a complete uniform space such that its underlying top olog-

ical space (X; X ) is fully normal. Put J =[V ]. Since (X; V ) is complete, =

V X

J q

X X

and since (X; X ) is fully normal, = = is a uniform structure in the sense

J q V

X X

of Tukey.Furthermore, (X; J ) is regular. Thus, (X; J ) is fully normal.

X X

\=)". Let (X; V ) b e a complete uniform space such that (X; [V ]) is fully normal. Put

J =[V ]. By assumption, (X; ) is uniform. Since = ,(X; X ) is fully

X V

J J V

X X normal.

HIGHER SEPARATION AXIOMS 331

4.14 Remarks. 1) By 4.13., the uniform space IR of real numb ers is paracompact as a

semiuniform convergence space. Furthermore, the top ological space IR of real numb ers is

paracompact as a semiuniform convergence space (cf. 4.6.).

2) A complete uniform space need not have an underlying top ological space which is fully

normal as the example under 3.6.2) shows.

3) The underlying Kent convergencespace of a ful ly normal semiuniform convergence

spaceneed not be ful ly normal; namely, there is a separated proximitiy space whose un-

derlying top ological space is not paracompact: Consider a completely regular Hausdor

space which is not paracompact (e.g. the ordinal space of all ordinals less than the rst

uncountable ordinal ! ). Continuing as under 3.6. 1) leads to the desired example.

4.15 Prop osition. Every subspace (in SUConv) of a ful ly normal (resp. paracompact)

semiuniform convergencespace is ful ly normal (resp. paracompact).

Pro of. Let (X; J ) b e a fully normal semiuniform convergence space, U X and J

X U

the initial SUConv{structure w.r.t. the inclusion map i : U ! X :

a) (U; J ) is regular since (X; J ) is regular (cf. [15; 5.3.]).

U X

b) (U; ) is a subspace (in Fil)of(X; ). Since subspaces in Fil ( Fil{Mer) are

J J

U X

formed as in Mer,(U; ) is a subspace (in Mer)of(X; ). Since the construct

J J

U X

Unif of uniform spaces (and uniformly continuous maps) is bire ectivein Mer,(U; )

is uniform, i.e. (U; )ism{uniform.

It follows from a) and b) that (U; J ) is fully normal.

Since obviously subspaces of T {spaces are T {spaces, the ab ove prop osition is also valid

1 1

for paracompactness.

4.16 Remarks. 1) As is well{known subspaces in Top (resp. Top ) of paracompact

top ological spaces need not b e paracompact. But if subspaces are formed in SUConv

one obtains from 4.15. the following result: Subspaces of paracompact topological spaces are

paracompact .

2) A paracompact semiuniform convergence space need not b e a subspace (in SUConv)

of a paracompact top ological space; namely the example under 3.8.2) is a counterexam-

ple (indeed, (X; J ) is paracompact!). Together with 4.15. one obtains that the class of

semiuniform convergence spaces which are subspaces of paracompact top ological spaces is

a prop er sub class of the class of paracompact semiuniform convergence spaces.

3) Since paracompact ness is not nitely pro ductivein Top ,itis also not nitely pro-

ductive in SUConv (cf. the corresp onding result for normality under 3.8.3)).

4) Full normality is related to the other higher separation axioms by means of the fol-

lowing implication scheme:

proximity

=) fully normal =) normal =) comp etely regular =) regular

space

uniform space

Adding the T {axiom one obtains from the ab ove implication scheme the following one: 1

332 GERHARD PREU

separated

proximity

=) paracompact =) T =) T =) T =) T =) T

4 3 2 1

space

separated

> uniform space

5. Dimension functions

5.1 De nitions. A) Let (X; ) b e a merotopic space.

1) The large dimension Dim (X; )of(X; ) issaidtobe n provided that every uniform

cover U of X has a re nement V2 of order n + 1 (i.e. each x 2 X is contained in at

most n + 1 elements of V ). The precise number Dim (X; ) is the smallest such n,or1

for the sp ecial case that X is empty; and wewrite Dim (X; )=1 if there is no such n.

2) The smal l dimension dim (X; )of(X; ) is de ned to b e the large dimension of

(X; ). Esp ecially, dim (X; ) n i every nite uniform cover U of X has a ( nite)

re nement V2 of order n +1.

B) Let (X; J ) b e a semiuniform convergence space.

1) a) The large lter{dimension Dim (X; J ) of (X; J ) is de ned to be

f X X

Dim (X; ).

b) The smal l lter{dimension dim (X; J )of(X; J ) is de ned to b e dim (X; ).

f X X

2) a) The large uniform dimension Dim (X; J )of(X; J ) is de ned to b e the large

u X X

dimension of (X; ) where (X; [V ]) denotes the underlying uniform space of (X; J ).

V X

b) The smal l uniform dimension dim (X; J )of(X; J ) is de ned to b e the small dimen-

u X X

sion of (X; ) (cf. a)).

5.2 Remarks. 1) a) For symmetric top ological spaces (X; X ), one writes

dim (X; X ) instead of dim (X; ) (resp. Dim (X; X ) instead of Dim (X; )), where

X X

(X; ) denotes the merotopic space correp onding to (X; X ). Then dim (X; X ) coincides

with the (Leb esgue) covering dimension of (X; X ) (cf. [12; de nition I.4])

b) For paracompact top ological spaces (X; X ), dim (X; X )=Dim(X; X ) (cf. [11; 9{14]).

2) a) Let (X; V ) b e a uniform space. Then dim (X; )= dim (X; [V ]) coincides with

V u

Isb ell's uniform dimension d (X; ) and Dim (X; )=Dim (X; [V ]) is identical with

V V u

Isb ell's large dimension d (X; ) (cf. [9]).

b) dim (X; [V ]) = Dim (X; [V ]) provided that Dim (X; [V ]) < 1 (cf. [9]).

u u u

5.3 Prop osition. 1) If (X; X ) is a symmetric topological space, then dim (X; J )=

dim (X; X ) and Dim (X; J )= Dim(X; X ).

2) If (X; X ) is a paracompact topological space, then additional ly, dim (X; J )=

dim (X; J ) = Dim (X; J ) = Dim (X; J ).

f u f

q q q

X X X

), where consists of Pro of. 1) (X; ) is the underlying lter space of (X; J

q q

X X q

all convergent lters on (X; X ). Hence, (X; )= (X; ), which implies the desired

equalities.

2) Since (X; X ) is paracompact, there is a nest uniformity V on X which induces X and it

is easily checked that (X; [V ]) is the underlying uniform space of (X; J ). Then = .

V X

)= dim(X; ) = dim (X; )= This implies (together with 1)) that dim (X; J

V X u

q X

HIGHER SEPARATION AXIOMS 333

dim (X; J ) = dim (X; X ) and Dim (X; J ) = Dim (X; ) = Dim (X; ) =

f u V X

q q

X X

Dim (X; J )=Dim(X; X ). Consequently, the desired result follows from 5.2.1) b).

5.4 Remark. For normal symmetric top ological spaces (X; X ), Dim (X; J )may dif-

fer from dim (X; J ), e.g. if (X; X )= , then dim = 0 and Dim = 1 (cf. [8;

f 0 0 0

5.4.8]).

5.5 Prop osition. Let (X; V ) beaproximity space (= total ly bounded uniform space).

Then

dim (X; [V ]) = dim (X; [V ]) = dim (X; )= Dim(X; [V ]) =

u f V

= Dim (X; [V ]) = Dim (X; ).

f V

Pro of. Since = (cf. (2) in the pro of of 4.11), dim (X; [V ]) = dim (X; )=

V u V

[V ]

dim (X; )= dim (X; [V ]) and Dim (X; [V ]) = Dim (X; ) = Dim (X; [V ]). Further-

f u V f

[V ]

more, Dim (X; ) = dim (X; ) b ecause ( ) = .

V V V c V

5.6 Remark. For a given proximity space the common value of all dimension functions

considered ab ove is known to b e Smirnov's {dimension of it (cf. [20]).

5.7 Prop osition. If (X; V ) is a complete uniform space, then dim (X; [V ]) = dim (X; X )

f V

and Dim (X; [V ]) = Dim (X; X ), where (X; X ) denotes the underlying topological space

f V V

of (X; V ).

Pro of. Since by assumption = , the desired result is obvious.

[V ]

n n n n n n

5.8 Corollary. Dim IR =DimIR = Dim IR =Dim IR =DimIR =dimIR =

u f f u

t t t u u t

n n n n

dim IR = dim IR =dim IR =dim IR = n.

f u f u

t t u u

is a paracompact top ological space, it follows from 5.3. that Pro of. Since IR

n n n n n n

, =Dim IR =Dim IR =Dim IR = dim IR = dim IR dim IR

f u f u

t t t t t t

n n

where dim IR = n (cf. e.g. [4; 7.3.19]). Since IR is a complete uniform space, it follows

t u

n n n n

from 5.7. that dim IR =dim IR and Dim IR = Dim IR .

f f

u t u t

n n

Furthermore, dim IR = Dim IR = n (cf. [9; V.12]). Thus, the corollary is proved.

u u

5.9 Prop osition. If (X; J ) is a semiuniform convergencespace, then the fol lowing are

valid:

a) dim (X; J ) Dim (X; J ),

f X f X

b) dim (X; J ) Dim (X; J ).

u X u X

Pro of. a) and b) follow immediately from

dim (X; ) Dim (X; )

for each merotopic space (X; ) (cf. [8; 5.4.7]).

5.10 Prop osition. 1) Let (X; J ) be a semiuniform convergencespaceand(A; J ) a

X A

334 GERHARD PREU

subspace (in SUConv) of (X; J ). Then the fol lowing aresatis ed:

a) Dim (A; J ) Dim (X; J ),

f A f X

b) dim (A; J ) dim (X; J ).

f A f X

2) Let (X; J ) be a principal uniform limit space (= uniform space) and (A; J ) a

X A

subspace (in SUConv) of (X; J ). Then the fol lowing aresatis ed:

a) Dim (A; J ) Dim (X; J ),

u A u X

b) dim (A; J ) dim (X; J ).

u A u X

Pro of. 1) and 2) follow immediately from the fact that for each merotopic space (X; )

and each subspace (A; ) (in Mer)of(X; ) the following are valid:

a) Dim (A; ) Dim (X; ),

b) dim (A; ) dim (X; )

(cf. [8; 5.4.10]).

5.11 Corollary. Let X be a symmetric topological spaceandA a closed subspaceinTop

(resp. Top ). Then

dim A dim X and Dim A Dim X.

Pro of. Since closed subspaces in Top are formed as in SUConv the desired result

follows from 5.10.1) and 5.3.1).

5.12 Prop osition. 1) Let (X; X ) bearegular topological spaceand(A; J ) a dense

).Then subspace (in SUConv) of (X; J

Dim (X; J ) = Dim (A; J ) :

f f A

2) Let (X; J ) be a principal uniform limit space (= uniform space) and (A; J ) a dense

X A

subspace (in SUConv).Then

Dim (X; J ) = Dim (A; J ).

u X u A

Pro of. 1) and 2) follow from the fact that for each regular merotopic space (X; )and

each dense subspace (A; )of(X; ), Dim (A; )= Dim(X; ) (cf. [8; 5.4.11.]).

A A

5.13 Theorem. 1) Let (X; ) be a normal lter space. Then dim (X; J ) n i every

Cauchy continuous map of any subspace (A; )(in Fil) of (X; ) into the n{sphere S has

a Cauchy continuous extension over (X; ), where S is endowed with its usual topological

Fil{structureconsisting of al l convergent lters w.r.t. the usual topology of S .

2) Let (X; J ) be a principal uniform limit space (= uniform space). Then dim (X; J )

X u X

n i every uniformly continuous map of any subspace (A; J )(in SUConv) of (X; J ) into

A X

n n

the n{sphere S has a uniformly continuous extension over (X; J ),where S is endowed

with its usual uniform structure.

Pro of. For every normal lter space (X; ), (X; ) is a normal nearness space, and for

every uniform space (X; [V ]), (X; ) is a normal nearness space. Thus, the ab ove theorem

follows from [13; 7.2.17.] (note that the corresp onding merotopic structure of the usual

top ological Fil{structure of S is the usual uniform structure).

HIGHER SEPARATION AXIOMS 335

5.14 Corollary. Let (X; X ) beanormal R {space. Then dim (X; X ) n i every

continuous map of any closed subspace (A; X )(in Top) of (X; X ) into the n{sphere S

has a continuous extension over (X; X ).

5.15 Remark. For normal Fil{determined semiuniform convergence spaces (X; J )

of nite small lter dimension dim (X; J ) = dim (X; ), dim (X; J )canbechar-

f X f X

acterized cohomologically (cf. [13; 7.3.3.] and note that a pair of lter spaces is a pair

(X; A), where X =(X; ) is a lter space and A =(A; ) is a subspace (in Fil)ofX,

n n

)). Similarly, for uniform spaces (= and H (X; A) is de ned to b e H ((X; ); (A;

A f f

principal uniform limit spaces) (X; [V ]) of nite large uniform dimension Dim (X; [V ]) =

Dim (X; ), Dim (X; [V ]) can b e characterized cohomologically (cf. [13; 7.3.3.] and note

V u

that for pairs of uniform spaces ((X; V ); (A; V )), H ((X; V ); (X; V )) is de ned to b e

A A

H ((X; ); (A; ))).

V V

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Department of Mathematics, Free University of Berlin, Germany