Topologies between compact and uniform convergence on function spaces

Internat.

VOL. 16 NO. (1993) I01-II0

TOPOLOGIES

BETWEEN COMPACT AND

UNIFORM

CONVERGENCE

ON FUNCTION SPACES

S.KUNDU

Divisionof Theoretical Statistics and Mathematics Indian StatisticalInstitute

203

B.T. Road,

Calcutta700 035 India

and

R.A.McCOY

Department

of Mathematics

VirginiaPolytechnic Institute

&

State University

460McBryde Hall Blacksburg,

VA

24061-0123

U.S.A.

(Received June

27, 1991andrevisedformJuly 31,

1991)

ABSTRACT. Thispaper studies two topologieson theset of all continuous real-valuedfunctions

on a Tychonoff space which lie between the topologies of compact convergence and uniform convergence.

KEY

WORDS AND PHRASES.

Function spaces,

Compact-open

topology, Topology of uniform convergence, a-compact,Pseudocompact.

1991

AMS

SUBJECT

CLASSIFICATION CODES.

54C35,54D45. 0.

INTRODUCTION.

The set C(X) of all continuous real-valued functions ona Tychonoffspace Xhasanumber of natural topologies.

Two

commonlyusedtopologiesarethecompact-open

,topology

and thetopology

of uniform convergence. The latter topologyhas been used formore thana centuryas the proper

setting tostudy uniform convergence ofsequencesof functions. Thecompact-open topology made

its appearance in 1945 in a paper by Fox

[3],

and soon after was developed by Arens in

[1]

and

Arens and Dugundji in

[2].

This topology was shown in

[6]

to be the proper setting to study

sequences of functions which converge uniformly on compact subsets. One of the distinguishing

features of thistopologyis thatwhenever

x

islocally compact thecompact-open topologyonC(X)

isthe coarsesttopology makingtheevaluationmape:X C(X)--.continuous

(where

e(x,])=

f(x)).

The compact-opentopologyandthetopologyofuniformconvergenceareequal ifand onlyif

x

is compact.

Because

compactness is such a strong condition, there is a considerable gap between these two topologies. This gap was especially felt in

[8]

while studying the completeness of a

on

C*(X)

whichwasstudied in

[5].

Thepurposeof thispresent workis toextend theseideasto two natural topologiesonC(X), to study the properties of thesetopologies, and to relate thesetopologies tothecompact-opentopologyandthetopologyof uniformconvergence.

Let Ck(X and_Cu(X denoteC’(X)withthecompact-open topology andthetopologyof uniform convergence, respectively. The definitions of the other twotopologiesthatwestudy here arebased

on thefact that Ok(x) canbe viewed in twodifferent ways. First wecan viewthe compact-open

topologyas a

"set-open

topology,"whereasubbasic open set looks like [K,

Vl

{I ec(x):l(K)c_ v},

whereK isa compact subset ofX andVis anopeu subset of the reals. Notethat

v

can alwaysbe takeuas abouuded opeu iuterval. Thesecoud waythatwe canviewthecompact-opeu topologyis

as a"uniformtopology." Thisapproachisdevelopedinthe nextsection.

Throughout the rest ofthe paper, we usethefollowingconventions. Allspaces areTychonoff

spaces. If

x

and Yareany two spaceswiththesameunderlying set,thenweuseX Y, X_<I/and X<Y to indicate, respectively, that X and Y have the same topology, that the topologyon Y is finer than or equal to the topology on X, and that the topology on Y is strictly finer than the

topology on

x.

The symbols t and 1 denote the spaces of real number and natural numbers, respectively. Finally, the constant zero-function inC(X)isdenotedby

Jr0.

1.

TOPOLOGIES OF UNIFORM CONVERGENCE.

In

thisfirstsectionwelook at "uniformtopologies" onC(X)inageneralsetting. This isdone in terms ofcertain pseudo-seminorms on C(x).

By

a pseudo-seminormon a real linear space g_is

meant areal-valued functionponEsuchthat

(1)

p(O) 0,

(2)

p(z) p( z) for allz(E, and

(3)

p(=

+

y)<_p(z)

+

p(y)for all =, y(

e.

Notethatitimmediately follows thatp(z)>_0and p(.) p(y) <_p(. y)for all., y g.

A

pseudo-seminormpiscalledaseminormif thefollowingadditional condition holds.

(4)

p(t.) tip(*)forallm Eand

.

Ofcourseaseminormpisanormif p(z) 0wheneverz 0.

Letabe any family of subsets of

x

satisfying thecondition:

if A,B a, thenthereexistsaC asuch thatAoBC_C.

For

eachA c,andf>0, let

A,

{(I,9)eC(X)xC(X):lf(z)-9(z)l <efor all

z

A}.

Then it can be easily verified that thecollection {Ae:A c,, >0} isabase forsomeuniformity on

C(X). We denote the space C(X) with thetopology inducedbythis uniformity be

Co,

u(X). This

topologyiscalled thetopologyofuniformconvergenceon

.

Foreach

Jr

C(X),A aand >0,let

<f, A, e> {g {/C(X):If(z)-g(z)[ < for all A}.

Thenfor each

Jr

(C(X), the collection < Jr, A, >:A(/a, >O} forms aneighborhood base atjrin

C,,,,,(X).

Since the topology comes from a uniformity, then

C,,,,(X)

is completely regular. Ifa covers

x,

then

C,,

(x) is a Tychonoff space. When a={x}, we get the topology of unilorm

Nowfor eachA15a,define _{the pseudo-seminorm PA}onC(X)by

Also foreach At5 a ande>O,let

Let

PA(f) min{1, sup{ 1(*) :x15A}}.

VA,

{f15C(X):PA(I)< }"

{VA,:A

qt,e> 0}.

It can be easily shown that for each 115C(X),l

+

’r _{l

+

v:v

’} forms aneighborhood baseat

I.

Wesaythat thistopologyisgenerated bythecollectionof pseudo-seminorms {i0A:A15c,}. Notethat

ifwechoose0< <1,then for each115C(X),wehave

f

+VA,

_

<f,A,e> and

<I,A,>

C_I+VA,

for allA15

.

Thisshows that thetopologyofuuiformcouvergenceonais thesame asthetopology generated by thecollectiou of pseudo-semiuorms {pA:A15a}.

We

seefrom this poiut ofview that

C,,,u(x

isatopological group with respect to additiou.

Observe thatfor

C*(X)

wecanactuallyusetheseminorms_{PA defined}by PA(I) sup{ 1(.) :*15A}.

Cousequeutly, C*

,,,,(x)

is a locally couvex topological vector space. Oue might wouder wheu

C,u(X)

is a topologicalvector space. This is answeredbythe following theorem.

In

thistheorem the term "bounded" refers to a subset ofa space such that each restriction ofa continuous real-valuedfunctionon the space to thissubsetisaboundedfunction.

THEOREM 1.1. Thespace

C,,,,,(X)

isalineartopologicalspace if andonlyifeveryelement of

aisbounded.

PROOF.

Ifeveryelement ofa is

bounded,

thenasinthecaseofC* .(x),wecanactuallyuse

seminorms_{for PA when}Aea.

For the converse, suppose that there is an Aea which is not bounded. Then there is an

115C(X) and

{r.:n15N} _c

A such that _{l(r.)_>n} for each n. To show that scalar multiplication

cannot be continuous in

C,,,,,(X),

let T be the scalarmultiplicationoperator definedby T(t,g))=tg

for 15 and#e

C,.(X).

Weshow thatTisnotcontinuousat (0,l). Let

VA,,

beaneighborhoodof

lo

T((O,I)) in

C,,,,(X),

where A15a and 0< <1. Then for any neighborhood U of0 in

,

there existsanhensuchthat

neU.

But

xnI(..)

>1>, sothat

T((n,l))=I

qtVa,

.

As

a consequence of Theorem 1.1, if every element of a is pseudocompact

(a

space is

pseudocompact ifevery continuousreal-valuedfunctionon the spaceis

bounded),

then

C,,,,,(X)

isa

locallyconvextopological vectorspace. Conversely,if everyelementofa isC-embeddedin

x (that

is, every continuous real-valued functionon the subspace has a continuous extension to

X)

andif

C,,,,(X)

isatopologicalvector space, theneveryelement ofaispseudocompact.

For the restofthepaper,we areinterested inthe particular topology where a is theset of

a-compact

(countable

union of

compact)

subsets ofX.

In

this case, wedenote the space

C,,,,(X)

by

sametopologyifwetake the members ofr,tobetheclosures ofther-compact subsets ofX.

2.

THE

a-COMPACT-OPEN TOPOLOGY.

Forany subset A ofXandany open subsetVof

,

define [A, V] {f(C(X): I(A)C_V}.

This agrees with the usual

"set-open"

terminology for the compact-open topology because A is compact in this case. Nowlet ,(X) be theset of ,-compact subsets ofX, and let bethe set of bounded open intervals in

.

For the g-compact-open topoloev on C(X), we take as subbase, the family

{[A,B]:AG.(X),B };

andwedenote this spaceby Co(X). Notethat the sametopologyisobtained by using

[,B],

where A6,(X) and B(9. This is becausefor each f

.

C(X),

f(7t c_

f(A); so that

f(Tt

f(A). The fact thatCa(X)isaTychonoffspacecanbeprovedinamannersimilar tothe proof ofLemma5.1 in

[9].

It

isuseful to relatebasic open setsinCo(X to basic opensetsin

C,,,,,(X),

whichis donein the followingtwolemmas.

LEMMA

2.1. Let W _{f’__[ai,}v,]bea basicneighborhoodof

I

inC,,(x). Then thereexists

an >0such that <f,A O o

A,,

> C_ W.

PROOF.

For

each i, sinceI(A,) iscompact, thereis _{an ei}>0such that the ei-neighborhoodof

I(A,)iscontainedin

V,.

Theconclusionnowfollowsby taking

LEMMA

2.2. LetA(,(X), and let

I

(C(X)be such that I(A)is bounded. Then for each >0,

thereexists abasic open set W inC,,(X)suchthat_1"(W

c_

<)’,A,e >.

PROOF. Since I(A) is bounded, there are open intervals

v

v,

of length such that

I(A---

C:_ O,"__lV,. Foreachi, letA Ar3

f-(V

i), and let W be the

-neighborhood

ofV in

.

Then

each

Ai

r(X), and each

W,

is an open interval of

length

e. Sodefine

w

c,"=

[A,,W,], whichis basicneighborhoodoffin Co(X). Itisstraightforwardtocheck that

w

_

<f,A,e>.

Lemma

2.1 tellsus that the a-compact-opentopologyis coarser than orequal tothetopology

of uniform convergence on.-compact sets.

On

the otherhand, it isclearfrom definitions that the .-compact-open topology is finer thanor equalto the compact-open topology. Therefore wehave thefollowing generalcomparisons.

THEOREM2.3. Foreveryspace X,Ct(X <_ Co(X <

Co,

u(X < C,,(X).

It

is well-known that

C*(X)

is dense in

C(X).

Thesame is true for

"Co(X

), as we see in the nextresult.

THEOREM2.4.

For

everyspace

X,C*(X)

isdenseinCo(X).

PROOF. If

n’=

_I[A,,V,] isabasic opensetinCo(X containingf,then

o,=

iv,iscontained in

someinterval(a,b). Then if is definedbyg(z) f(z) ifa_<f(z)_<b,g(z) aif/(z)<aand (.) Iiif

b</’(z), wehave that

c*(x)

a r

:’=

_{[A,, V])}

asdesired.

Even though

C*(X)

isdensein_Co(X),the members of

C*(X)

playaspecial roleinCo(X inthe following sense. For each

I

.C(X), let

TI:Co(X)--,Ca(X

be the translation operator defined by

Tt(g

J’

+

gfor all g Co(X).

BETWEEN ND UNIFOR CONVERGENCE 105

PROOF. Forthe sufficiency, supposethatlE

C*(X)

and that [A,V]isasubbasicneighborhood of

Tl(g)

in_{C,,,(X) for}some ge C,,(X). Then by

Lemma

2.1, thereexistsan >0such that

<

TI(g),A,

> c_[A,V].

Now (f+g)(A)=

TI(g)(A

is bounded. Since fE

C*(X),

g(A)=(f +g)(A)-f(A) is bounded. So by Lemma2.2, thereexists abasic open set

w

inC,,(X)suchthatge

w

c_ <g,A,>. Weseethat

T.(W)

C_Tj(< a,A,> C_ <Tj(a),A,> C_[A,V], and thus_{TI must}becontinuous atg.

For the necessity, suppose that j"

.C*(X).

Then

I

is unbounded on some Ar(X). Now

[A,(-1,1)] is aneighborhood of

f0

Tl(-.f)

in C,,,(X).

Suppose,

bywayof contradiction, that T₁is continuous at -l. Then there would be a _{basic neighborhood n,"__l[A,,Vi] of --f} in C,,(X) such that

TI( 1"I=

I[A,, V])

C’_[A,{-I,I)].

Since

-I

would be bounded on

t.J’=:A,,

so would

I.

Therefore there would exist some

zeA\t",=:A,. Then let eC(X) be such that (z)= and (u)=0 for all

re

t-J’=iA,, and define

h=b-f. Clearlyhe f,"-_I[A,,V,], sothat

l+h=Tl(h)e[A,(-l,1)].

Butthen =(f+h)(z)e(-1,1), which isacontradiction. Therefore

T!

isnot continuous at _-f.

COROLLARY2.6. Thefollowingareequivalent.

(1)

C,,(x)isatopological vector space.

(2)

C,,(x)isatopological semigroupunder addition.

(3)

c(x) c;(x).

(4)

Xispseudocompact.

When studying function spaces, it is sometimes useful to use induced functions. That is, if 4:XY is a continuous function, define

b*:C(Y)C(X)

by

*(#)=#o

for all eC(Y). We can

establishthefollowingtheorem muchlikethecorrespondingtheorem for the compact-opentopology

(cf.

[10]).

In

this

theorem,

the definition of

:X-Y

being a or-compact-covering mapis that each

a-compact subset of is contained inthe image ofsome#-compactsubset of

x

under

.

THEOREM2.7. If:X-Yiscontinuous, then

*:C,(Y)-C(X)

is continuous. Furthermore,if is ar-compact-coveringmap,then

*

isanembedding.

3.

COMPARISON

OF

TOPOLOGIES.

We

havealreadyseenthat

c,(x)<c(x) <

c,.(x)

< c.(x).

In

this section we determine when then inequalities are equalities and giveexamples to illustrate the differences.

THEOREM 3.1. Forevery space

x,

ct:(X C,,(x)if andonlyif the closure ofeacha-compact

subset of

x

iscompact.

PROOF. The sufficiency is straightforward.

For

the necessity, suppose that Ct:(x =Ca(X

and letA a(X). Then thereexistacompactset K in Xandanopenset V in such that

fo

e[K,V]_C[A,$\{1}].

Suppose,

bywayof contradiction, that thereexistsanz \K. Then there wouldbesomef C(X)

suchthat f(z)= andf(y)=0forally K.

But

then

f e[K,V]\[A,$\{1}],

COROLLARY

3.2. If C.(X)= C,,(X), then X iscountablycompact.

THEOREM3.3.

For

everyspaceX,C,,(x)=

C,,,,,(x)

if andonlyifX ispseudocompact. PROOF. If

x

is pseudocompact, then for each A ca(X) and

"

e.C(X), I(A) is bounded. Thereforeby Lemma 2.2, wehave thatC,,(x)=

C,,,(x).

For

the converse, suppose that

x

is not pseudocompact. Then there is some f

.

C(X) and

A a(X) such that I(A) is unbounded.

Now

for any >0, < I,A, > cannot beaneighborhoodof

!

in C,,(x). This is because for arbitraryopen W in C,,(x)there is some ge

w

oC*(x)

by Theorem

2.4; butg < I’,A,e>. Itfollows thatC,,(X) <

C,,,,,(x).

COROLLARY

3.4. If everya-compactsubset ofXhascompactclosure,thenCk(X)

C,,,,(X).

THEOREM 3.5.

For

every space

X,C,,,,,(X)=C,,(X)

if and only if X contains a dense

a-compact subset.

PROOF.

Suppose

that

x

contains an Aea(X) which is dense. Then for each

y

(;C(X) and >0,

<y,A,>

c_

<l, X, e>.

Thus < I,X,e > isaneighborhoodof

I

in

C,,,,(X),

sothat

C,,,,,(X)=

C,,(X).

Conversely,let

C,(x)

C,,(x), and consider < I0,X,e > where0<e<1. Then thereexists an Aea(X) and6>0suchthat

<Y0,A,6>

c_

< 10,X,e >,

Suppose,

by way ofcontradiction, that there is an xe

X\].

Then there would be some

!

.

C(X) with I(x) and l(v) 0for allv e].

But

thiswouldmeanthat

Y

-

< fo,A,6> \ < fo,X,e> which isacontradiction. Itfollows thatX ].

COROLLARY

3.6. IfX isseparable,then

C,,,,,(X)=

C,,(X).

We

end this section by looking at some examples which illustrate all possible inequalities

betweenthese functionspaces.

EXAMPLE

3.7. If

x

isanycompactspace, then

c(x) c(x)

c,,,,,(x)

c,,(x).

EXAMPLE

3.8. If

x

isanuncountablediscretespace,then C,(x)<C(X)<

C,,,,,(x)

< C,,(x).

EXAMPLE

3.9. IfX hiorX

,

then

c(x)<c.(x) <

c.,.(x)

c.(x).

EXAMPLE

3.10. LetXbethe space ofcountableordinals withtheordertopology. Then each a-compact subset of X has compact

closure,

but X does not contain a dense a-compact subset. Therefore

C,(X) CAX)

C,,,(X)

<C,,(X).

EXAMPLE

3.11. Lethi* fhi\hi, wherefhiisthe

Stone-ech

compactificationofhi. Let1bea

point of hi* whichisnot aP-point

(cf.

[11]).

DefineXto behi*\{p}. ThenX iscountably compact,

but doesnothave adensea-compactsubset

(since

pointsarenot

G6-sets

).

Alsosincepisnot a

BETWEEN CONVERGENCE 107

EXAMPLE

3.12. Let X=\{p}, where peN*

(see

Example

3.11).

compact andseparable, but notcompact. Therefore

Then X is countably

Ck(X)<Co(X)

Co,

.(x) C.(X).

Notethatthisshowsthat theconverseofCorollary3.2isnot true.

An

exampleofaspacewhich is notcountably compact but whichhas thesesamefunction space relations isthespaceq in5I of

[4].

4.

ADDITIONAL

PROPERTIES.

Theorem 3.5 can be expanded to include some additional properties of the spaces Co(X and

co,

.(x).

THEOREM4.1. Thefollowingareequivalent.

(I) Co,.(X

is (completely)metrizable.

{2)

Co,.(X

isfirst countable.

(3)

{lo}isa

Gs-set

in

Co,,,(X).

(4)

Co(X)issubmetrizable.

(5)

Co(X)isfirstcountable.

(6)

{Io}is a

Gs-set

inCo(X).

(7)

Xcontainsadense a-compact subset.

PROOF.

That

(7)

implies

(1)

follows from Theorem 3.5 since C,,(x) is always completely

metrizable.

Toprove that

(3)

implies

(7),

suppose that X doesnot containadense a-compact subset. Now

a

Gn-set

is a subset which canbe written as acountable intersectionof open subsets. Soforeach

n6N, let <f0,A,,,e,> be any basic neighborhood of

Y0

in

Co,,,(X

). We know that there is some z fiX\

tJ=

A,.

Let f6C(X) be such that f(z) andf(y) 0for ally

J.=A..

Then

f

Nn=

<fo,

An,

6n>"

Butsincef #f0, weseethat{f0}isnota

G6-set

in

Co,

u(X).

Since

(6)

implies

(3),

it remains only to show that

(7)

implies

(4)

and

(5).

Suppose

that X contains a dense a-compact subset D. Let

I

beany element ofCa(X). We need to demonstrate that

I

hasacountable baseinCa(X). Wemaytake for$thefamilyofboundedopenintervalsin l withrational endpoints. Also letCbe theclosures of the members of05. Thendefine

.a

{.f-(C)oD:

e}.

SinceD is a-compact, atisacountable subfamily of a(X). Since

*

isalsocountable,itnowsuffices to show thatfor each subbasic neighborhood [$,v] of

!

in Co(X),thereis an A6at andB6

*

such

that/" [A,B]

c_

[S,V].

If

!

6[$,V],thenI($)

c_

V.

Hence

since iscompact, thereexist C6

e

andB6 such that

Therefore ifwedefine A

y-(C)Cl

D, thenwehave

f(A)

_

f(f-

I(C))CI

f(D) C_ CCIf(D)

_

B.

This means that f

.

[A,B]. To show that [A,B]C_[S,V], let ge;[A,B] and let

-’.

If W is a

neighborhoodoft, then

g-l(W)n

S $. Since

KUNDU AND R.A. McCOY

theng-l(W)nf-(intC)#. Butthen

g

(w)

n

! (i.t

c)nD#

.

so that

g-(w)na.

Therefore Wng(A), and thus t6-(A)CZ_B. It follows that g6-[$,V], so that

!

6-[A,B]

c__

IS,V]. Wecannowconclude that

!

hasacountable baseinGo(X).

Finally, to show that

(7)

implies

(4),

write the dense set D=

u=

A,,,

where each

A,,

is a

compact subset ofX.

A

submetrizable space isonewhichcanbemappedintoa metric spaceby a continuous injection.

(Note

thatsingleton sets are

Gn-sets

ina submetrizablespace.) Define Y to be the disjoint topological sum

=A,,

and let O:YX be the natural map. Thenthe induced function

4*:C,,(X)C,,(Y)

is continuousbyTheorem 2.7. Since 0(Y)isdense inx,*is aninjection.

Nowthesummap

s:c.(’=

A.)-[I’=

C.(A.)=

l-l=

is a continuous bijection, since it is a homeomorphism when defined on

Ck(’=

An)

(cf.

[6]).

ThereforeS

o*

is acontinuousinjection fromCa(X intothe metrizable space

l’I’=

Ck(A,,}, which makesCa(X)submetrizable.

Forourlast topic,weconsiderthe separability ofCa(X),and the weakerpropertyof having the

countable chain condition

(i.e.,

every pairwise disjoint family of nonempty open subsets is

countable).

We

seefrom thefollowing theorem and Theorem 3.3 that

C,u(X

is separable

(has

ccc)

if andonlyifCa(X isseparable

(has ccc).

THEOREM4.2. IfCa(X hasthe countablechaincondition, thenX ispseudocompact.

PROOF.

Suppose

that

x

is notpseudocompact. Then thereis aclosed C-embedding

:N-.X.

So by Theorem 2.7, theinduced function

*:C(X)-C(N)

is continuous. Since theembeddingisa

C-embedding, $*isasurjection.

We show that Ca(N does not have the countable chain condition. Then since

*

is a

continuous surjection, Ca(X will also not have the countable chain condition.

Let

U=(0,1) and

v

(1,2), and for each Ainthepower set9(N) of N, define

S_A=[A,U]O[N\A,V].

Then _{Sa is}anonempty basic open set in

C(). Suppose

that A,B6-@(NI)with A B. Without loss of generality,say thereissomez6-A\B. If

"

6-SA, then I(z)6-U; but if16-

SB,

then I(z)6-

v.

Since U V

,

SA

SB

.

Therefore

{SA:A

6.()} is all uncountable pairwise disjoint family of

nonemptyopen subsets ofCa(N).

To

completethe characterization ofC,(X) being separable,weneed thefollowing lemma.

It

is aconsequence of Theorem 8.17in

[4];

however,wegiveasimple directproof.

LEMMA

4.3.

Every

pseudocompact submetrizable

(Tychonoff)

space is compact and metrizable.

PROOF.

Suppose

topological space (X,r) is pseudocompact and has a metric d which

generatesastrictlycoarsertopologyr_don X. Thenthereexists anz 6-U6-r\rdand sequence(zn)of

distinct points of X\U such that (zn) converges to z in r_d. For each n, let B be an open

neighborhood of z,in r_d having diameter less than

1

and such that

BBI=$

for _ij.

Now

let V6- rbeaneighborhoodof whose

closure,

,

inriscontainedin U.

For

eachn,define

V,

B\,

which is a neighborhood of

,,

in r. Note that _{V,,:n6-N} is a discrete family in r.

For

each n,

choose] eC(X) such that f,()=nandf(y)=0for ally6-x\v,. Define_{f e}C(X)by y(y)=f(y) if

y 6-

V

andf(y) 0otherwise. Since

y

isunbounded,this contradicts (X, r)being pseudocompact.

TOPOLOGIES BETWEEN COMPACT AND UNIFORM CONVERGENCE 109

PROOF. If C,,(X)is separable, then the smallerspaceC,(X)is separable. This meansthat X is submetrizable. Theresultnowfollows from Theorem4.2andLemma4.3.

QUESTION

4.5. When is _C,,(x) metrizable?

In

particular, is C,(N) metrizable or even

normal?

ACKNOWLEDGEMENT.

Thefirst author’s researchis supported bytheCouncil of Scientific and Industrial Research,India.

REFERENCES

1.

ARENS,

R.,

A

topologyfor spaces of transformations, Annals ofMath.47

(1946),

480-495. 2.

ARENS,

R.

&

DUGUNDJI,

J.,

Topologiesforfunctionspaces, Pacific

J.

Math. 1

(1951),

5-31. 3.

FOX,

R.H.,

On topologiesfor functionspaces, Bull.

Amer.

Math.

Soc.

51

(1945),

429-432. 4.

GILLMAN,

L.

&

JERISON,

M.,

Rings ofcontinuous functions,

D.

Van Nostrand, Princeton,

New

Jersey,

1960.

5.

GULICK, D.,

The a-compact-opentopologyanditsrelatives, Math. Scand. 30

(1972),

159-176. 6.

JACKSON, J.R.,

Comparison of topologies on function spaces, Proc.

Amer.

Math. Soc. 3

(1952),

156-158.

7.

KUNDU,

S.,

Spaces

of continuous linear functionals: something old and something new,

Topology Proc. 14

(1989),

113-129.

KUNDU,

S., McCOY,

R.A., &

OKUYAMA,

A.,

Spaces

of continuous linear functionals on

Ck(X),

MathematicaJaponica 34

(1989),

775-787.

McCOY,

R.A. & NTANTU, I.,

Completeness properties of function spaces, Topology and

Appl.22

(1986),

191-206.

10.

McCOY,

R.A. & NTANTU, I.,

Topological properties of spaces of continuous functions,

Lecture Notes

inMathematics, 1315,Springer-Verlag, Berlin,1988.

11.

VAN

MILL, J.,

An

introduction to/w, Handbookof Set-theoretic Topology, North-Holland,