One point compactification on convergence spaces

Internat. J. Math. Sci. VOL. 17 NO. 2 (1994) 277-282

277

ONE-POINT

COMPACTIFICATION

ON

CONVERGENCE

SPACES

SHINGS.SO

CentralMissouriStateUniversity Warrensburg,Missouri

(Received December 22, 1992 and in revised form February 11, 1993)

ABSTRACT.

A

convergencespace isasettogetherwithanotionof convergence ofnets. Itiswell knowrhow the one-point compactificationcanbe constructedonnoncompact, locally compact topologicalspaces.

In

this paper,wediscusstheconstructionoftheone-point compactification

onnoncompactconvergencespacesandsomeofthe properties of the one-point compactification of convergencespacesarealsodiscussed.

AMS(MOS)

SUBJ.

CLASSFICATION:

Primary: 54A20, 54B20, 54D35.

KEYWORDS:

E-subnet, reversibleE-subnet, pseudotopological, pretopological,limitsuperior, limitinferior, powerset, compact, compactification.

1. Preliminaries.

The followingdefinitionswereintroducedin

[1]

inestablishingthe definitionof convergence spaces.

A

net is a map s whose domain is a directed set. Let s be a net with domain D. If

m

D,

then

s(m)

willbedenotedbys,. If

D

is directed by therelation

>

andm

D,

then

{n

Din

> m}

will be denoted by roD. If

E

C_

D,

then

E

is

cofinal

in

D

if for eachmin

E,

mD

f3

E

and

E

isresidualin

D

if for eachmin

E, mE

roD. Thedomainand range of

afunction

f

will bedenoted by

D(f)

and

R(f)

respectively. If

X

is aset ands isa net with domain

D,

then sis i__n_n

X

if

R(s)

C_

X.

Supposes is anet with domain

D

and isanet with domainE. Then tisasubnet of s, denoted byt

<

s, if for each n

D,

there exists m E

E

such that

t(mE)

C_

s(nD).

A

universal netis anetwhich hasno

poper

subnet.

A

convergencestructureontheset

X

isaclassCoforderedpairssuch that

(1)

if

(s,z)

C,

thensisanet in

X

andx

X,

and

(2)

if

(s, z)

Cand is a_{subnet of s, then}

(t, z)

_{C. If}

Cisa_{convergence structureon}

X,

_then

(X, C)

_orjust

X

iscalleda_{convergence space, and the}

statements_{convergesto x,denoted by s}

x,means

(s, x)

EC.

Furthermore,

X

is compactif every net inX hasaconvergent

subnet,

madXis

Hausdroff

_ifnonet in

X

convergestotwo distinct pointsofX.

Unlessit isspecified, Xwillbeused to denoteaconvergencespace

throughout

this paper. Thefollowing theoremis straightforward.

Next,

weintroduce (-subncts and reversibleE-subnetswhichareimportant inour discus-sion.

Let Sbeanet ofsets with domain D and let be anet with donainE. Tiler is called

an -subnetof

S,

denoted by

<e

S,

if foreach n(

D,

thereexists m

E

sothatfor each p

>_

m there is q

>_

n such that

t),

Sq.

Similarly, the notation S

<

will mean that for each m

E,

there exists n E

D

so that foreach q

>

n, there is p

>_

m such that

tq

(

S.

Furthermore, is calleda reversible -netofSif

_<

SandS

<

t.

Let

X

be a convergence space, and let S be a net of sets in X. Then the set of all x such that some -subnet ofS converges to x is called the limitsuperior of

S,

denoted by lira sup

S,

and theset of allx suchthatsomereversible (-subnet of5’converges tox iscalled thelimit

inferior

of

S,

denoted bylira in

f

S.

Z. Frolik

[2]

establishedthefollowingtheorem fortopologicalspacesandSo

[3]

showed that theresult also holds for convergencespaces.

THEOREM1.2. If

X

isaconvergencespace,Sisanetofsetssuch that for eachnfi

D(S)

S,

C_

X,

and

T

_<

S,

thenlira in

f

SC_ lira in

f

TC_ lira sup

T

C_ lira supS. Thenext lemma follows inunediatelyfrownthe definitions involved.

LEMMA

1.1.

Suppose

s is anet and

T

and Sare netofsets. Ifs

-<e

T

and

T

S,

then

s<eS.

LEMMA

1.2. Ifs

_<e

S,

thenthere isasubnet

U

ofSsuch that s isareversible (-subnet ofU.

PROOF.

Let

D

D(S)

and

E

D(s).

Let

F={(m,n)

eDxE [s,e

S,,,}

with the

crossproduct order. Then

F

is adirectedset.

Let U

be the net with domain Fdefincdby

U(m,n)

S,,. Let

p D.

Sinces_<e

S,

thereexistsq

e

E

such that every element of

s(qE)

is

belongs

tosomeelement of

S(pD). Let

(m,

n) >_

(p, q).

Thenwehavem

>_

p and

U(m,

n)

S,. Thus

V

_<

S.

Let

k

E. Sinces_<e

S,

thcrecxists

q>_

ksuchthat sq

U(p,q)

for somepinD. Let

(m,

n) _>

(p,

q).

Thenwehaven

>_

kand s,, E

U(m,

n).

Therefore

V

_<

s.

Let

(h,k)

F. Sinces

<:e

S,

there exists p Esuch that ifn

_>

p, thereism

>_

h such that s,,

U(m,n).

Let q Esuchthat q

>_

k andq

>_p.

Henceq kEsuchthat ifj

>_q,

there is

>_

h such that

s

U(i,j), i.e., ifj

>_

q, there is (i,j)

>_

(h,q)

>_

(h,k)

such that

s

U(i,

j). Therefores

_<

_e U.

THEOREM 1.3. IfSisuniversal,then lira supS lira in

f

S.

PROOF. Let x lira sup S. Then there exists an-subnet s ofS such that s ---,x.

By

Lemma1.2, S_<9 s. Therefores isareversibleE-subnetofSandhence x lira

inf

S. Since lira supSC_ lira in

f

S,

itfollows froxn Theorem 1.2 that lira supS lira in

f

S.

If

X

isaset ands is anetwith domain

D,

thensis in

X

if

R(s)

C_

X,

sis eventuallyin

X

ifforsome min

D,

s(mD)

C_

X,

andsisfrequentlyin

X

iffor eachm

D

s(mD)

X

ONE-POINT COMPACTIFICATION ON CONVERGENCE SPACES 279

PROOF. Suppose isfrequentlyin.4. Let

D

D(s).

Then for each in

D,

s(iD)NA

#

O.

Let m

e

D andE

{n

e ,nD[s,, e

s(mD)

A}.

Then Eis adirected set. Let u

alE,

and let E roD. Then thereexists j E E such that u(jE) C_

s(iD)

fqA and henceu(jE) C_

s(iD)

andu(jE)C_ A. Therefore u

<

and

,,

iseventuallyin A.

Suppose

some subnct

,,

ofs is event,rally in A.

Let

D

D(s),

E

D(u),

andn

D.

Then thereexists m G E such that

u(mE)

C_

s(nE).

Sinceu is eventually in

A,

thereexists

E

such that

u(iE)

C_ A. Let j

>_

m andj

>_

z. Thenu(jE) C_

s(nE)

A. Therefores is

frequentlyinA.

ThefollowinglemmaisProposition 3.3 of

[4].

LEMMA

1.4. Ifsisanet inthe set

X,

thensisuniversal ifandonlyif foreachsubset

Y

of

X,

s iseventuallyin

Y

oreventually in

X

Y.

2.

THE

POWER SET AND

THE ONE-POINT

COMPACTIFICATION

_OF

X

Let

PX

denote the powcr set ofX. If

X

isa convergencespace,

L

X,

and

S

isanet inPX, then the statement that S LinTOX means thatL=limsupS=liminfS.

It

followsfrom Theorem 1.2 that

T’-X

isa convergencespace.

In thissection,weinvestigate the convergence structure of the powerset,

7:’X,

ofa conver-gence spaceX and theconstructionof theone-point compactification X* ofX. Wethen show thatX* ishomeomorphictoasubspaceofT’X*.

It

shouldbenoted thatin

[5]

and

[6],

G.D.

Richardsonand

D.C. Kent

studied the one-point compaetification and the starcompactificationonconvergencespaces definedbyfilters. There

are somesimilaritiesbetween the constructionof the one-point compactifieation of convergence spaces definedbyfilters and convergence spaces defined. Oneof theessential differences isthat the convergence structuredefined byfiltershas the constantconvergence propertybuilt inthe definition,while inthe convergencestructre definedbynetsthe constantconvergence property

isnot a.ssumcd.

Let

f

bc amap from X toY. The statcmcnt

f

is continuou,smeansthat if,s --,xin

X,

then

f

os

f(x).

The statcncnt that

f

is a homcomorphism means that

f

is one-to-one, onto, continuous, and

f-

is continuous. X is said to be homeomorphic to

Y

if there is a

honacomorphismfrownXontoY.

THEOREM

2.1. If

X

is nausdorffand

X’

{{x}lz

6

X},

then

X

is homeomorphic to thcsubspacc

X

of7:’X.

PROOF.

Lct h, bcamap from

X

to

X’

such that for cachx

X, h(z)

{z}.

Thenit is obviousthat h isonc-to-oncand onto. Let s x. Thens isarevcrsible6-subnet ofhosand hcnccz lira

inf

hos. Since

h(x)

{x}, h.(z)

C_ lira

inf

hos.

Let

y 6lira supho

.

Then thcreexistsan &subnct of ho s such that y. Since

X

isHausdorffand

_<

s,x y and hence y

h(x).

Thereforclira sup ho s c:_

h(x)

C_ lint

inf

hos. It followsfromTheorem 1.2 that hos

h(s).

Let S

{x}

in

X’.

Then thereexists a reversible6-subnet u of

S

with domain

E

such that u---, x. Let v h,

-aoSandD

D(S).

Let

n

E.

Then thereexists

SO

Thcrcforcv

_<

u and 1,c,,,-,;

h-’

oS

h-’({x,}).

Sincc h is onc-to-onc and onto, andboth h md h

-

arc cont,inuos,X i l,’rllicto timsubspacc

X

ofX.

Let

X

beaconw.rgcnc.SlW

,.

IfI

K,

then 4is theset ofall pointsx suchthatsome net in

M

convcrgc to

..

As.t D ide.,

X

_ifD X.

A

compactif,cationofaconvergence spacc

X

is anorderedpair

(Y,

h)

,’1 tlt

"

is compact, his ahomcomorphisxn of

X

into

Y,

and

h(X)

is dense in Y.

Let

X

be a noncompact converg’tce Sl;tce ad let be a point

no

in X.

Le

X*

X

U

{}.

Viewing tlw rc,slt it Lcnla 1.3, we define the convergence in X* follows. Sul)oses isa net inX*. Let

.

inX* forsome x in

X

if mxdonlyifsis frequentlyin

X

and

sIX

,

and lets inX* ifand only ifnoubnetofsin

X

convergesin

X.

Suppose

s isauniversal net inX. Then by

Lemma

1.4, sis eventulyin

X

or

{}.

Ifs iseventuallyin

{}

or

siX

zfor any .: in

X,

thennosubne ofs in

X

convergesin

X

d hences

.

Otherwises x fir somex in

X

accordingtothe convergencedefinedonX*. Consequently,wehave the followingthcorcn.

THEOREM

2.2. If

X

is a xtl’Xill,;t’t, c,nvcrgcnccpacc, thenX* is a compactification ofX.

THEOREM

2.3. If

X

is a otiCOml,ct Hasdorff convergencespace, thenX* is homeo-,,orphito thc

,,bVc

{.,:

}. e

X

0

i,,

(X

}).

PROOF.

Let

X’=

{{x}].,:

X},

and let h’be the mapfi’o,nX* to

X’U

{0}

such that for each x ia

X, h’(x)

{x},

and

h’()

0.

Note that h’isan extensionofthehomeomorphism hin Theorem 2.1. It is clear that h is one-to-one and onto. Suppose s

.

Let u be an

-subnetofh

os.

Then

uJX

x foreach x in X. Thereforelira suph

os

.

Thus h is

strongly continuous. Let S beanet in

X’U {$}

with domain

D

d s h

-

o S.

Suppose

S

{x}

forsome xinX. Then thereexists areversible-subnetu ofSwith domMn

E

such that u x. Let n E. Then thereexistsn D suchthat ifp m, thereis q nsuch that

S,

9

u,

and hence

s

uq. Therefores

5

u d thuss x.

Hence

h’-l _oS

h’-’({x}).

Suppose

S

.

Then lira sup S lira

inf

S and hence no -subnet ofS converges in X. Now s h-I _{oS. Let v s. Thenv S and hence v does not converges inX.}

Therefores mad thus h

’-

oS

h’-(O).

Therefore h’ is a homeomorphism, d hence X*is homeomorphic to

X’

O

{$}.

A

net

of

nets with domain

D

is a net s with domMn

D

such that for each n

D,

s,, is a net.

Its

is a net of nets with domain

D

and for each n

D, D(sn)

Dn,

then the diagonal netgeneratedby s, denotedby

&s,

isthe net suchthat

D(t)

D

x

H{Dn[n

D}

with thecrossproductorder and foreach

(n,

f)

D(t),

t(n,

f)

s,(f(n)).

Thestatement that

X

ispseudotopological at x means that ifs is a net in

X

such that eachuniversalsubnet ofs convcrgcstox, then .s x. Thestatelnentthat

X

ispretopological

at x means that ifs is anct ofncts in

X

with domain D such that for each n

D,

Sn

--

X,

then As x.

ONE-POINT COMPACTIFICATION ON CONVERGENCE SPACES 281

PROOF. Let x E X* and h.t .s lc ;t _{net in X* such that every universal subnet of s}

convergestox inX* Supposex isinX. Let u bcauniversalsubnet ofs. Sinceu-x, uis

frequentlyin

X

and

ulX

x. By

Lemma

1.3, u is eventually in X. According to Theorem 1.4,s isfrequentlyin X. Let vbcaunivcrsdsubnct of

.s[X.

Then visauniversalsubnet ofs

and hcncc v x. Since

X

is pscudotopological,

s[X

x and hences x.

Suppose

x co.

Let beasubnetofs. Suppose convergestosomepoint y inX. Let u beauniversalsubnet oft. Sinceu x,

X

isHausdorff, u

<

t, and y, itfollows that y x. This contradictsthe suppositionthat x co. Thereforenosubnetofs convergesin

X

and hences

--

o0. ThusX*

is

pseudotopological.

Thefollowingthree lemmasarcTheorems 14, 4, and 8in

[1].

LEMMA

2.1.

In

order that the convergencespace

X

ispretopologicalat x it isnecessary and sufficient that ifs is auniversal net ofuniversal nets in

X

such that for each n in

D(s),

s,, x, then

As

x.

LEMMA

2.2. If is a net of nets with donain

D

and

<

As,

then there exist acofinal subset

E

of

D

and a net u of nets with domain

E

such that for each n in

E,

u,,

<

sn

and

Au<t.

LEMMA

2.3. Ifs is anet with domainD and is nnetofnetswith domainD such that foreachn E

D

t, iseventuallyin

s(D),

tlwn At

<

a.

Tim following example shows thatalthouglaa convergencespace ispretopological, its one-pointcompactificationmay not bepretopologcial.

EXAMPLE. Let

X

{1,2,

3,-.-}

and let s x if andonly if s iseventuallyin

{z}

for eachnet sinX andx 5X. Suppose uisanet of nets in

X

with domain

D

such that u, x for eachn D. Let v beanet with domainD such that v,, z foreachn D. By Lemma 2.3, Au

<

v, foreach n D. ThereforeAu --}xandhence

X

ispretopological.

Let

X

U

{co}

be the one-point compactification of

X,

andlet s be the net of nets with domain

N,

the set of naturalnumber,such that foreach nE

N,

the,domainof thenet s, isN also.

ifm<_n Ifnisodd,

defines,,(rn)=

(m+l)-n

_ifn>m.

Ifn is even, define

s,(m)

co forevery n N. Let be a net withdomain N andlet t,,

s,(n)

for each n

e

N. Then

<

A and t,,

{

ifnis odd Itfollows that

7/-,

oo

co ifn iseven.

since

fiX

isa subnet of such that

fiX

1. Therefore As

7/-,

co and hence the one-point compactificationof

X

is notpretopological.

THEOREM

2.5.

Let

X

be a

Hausdorff,

pretopologicalconvergence space, andX* be the one-point compactification ofX. Suppose for every net of nets s in X* with domain

D,

s satisfiesthecondition thatifforeachn

D,

s,, co,Ax co. ThenX* ispretopological.

PROOF.

Let s be a universal net of universal nets in X* with domain

D

such that for each n in

D an

x inX*. Supposex X. Thenfor each n E D some subnet

tn ofsn

is

eventuallyin

X

and hence

s,lX

x. Let u be the net with domain

D

such that foreach

S.S. SO

is 1)retopological. SinceX* is_COmlact and As is universal, As y forsome in X*.

rom

Lcmma2.5 and thefact that X* is Hausdroff, As x.

Supposex

.

By the assumptionAs

.

It follows frownLemma2.1 thatX* is

pretlmlogical.

The following theoremis stated in

[7],

aproofisgiven for thesakeof completeness. THEOREM2.6. If

X

is compact,pseudotopological,andHausdorff,then

X

is topologicM ifand only if

X

isrcgular.

PROOF.

Suppose X

is topological. Let sbe anetofnets in

X

with domain

D

such that

s

xandfor eachn in

D,

,,

p,,. Lct q be auniversalsubnct ofp win donain

E.

Then q yforsomeg

X

since

X

is compact. Fix m0 D. Since q p, thereexists n0

E

such

that foreach n0, thereis) m0 such thatq, =p. Let be anet with

domainn0E

such that for each m0E, t,

.

with3

m0E.

Thent, q, for each

moE.

Thus At y because

X

istopological. Since

X

isHaush,rff, x y and henceq x. Since every universM subnet ofpconvergestox andX ispscudotopological,p x and hence Xisregular.

Suppose X is regular. Letpbe anet with domMn D suchthat p x and let s beanet ofnetswith donain

D

such that foreachnG

D, sn

p. Letu be auniversalsubnet of Thenu yforsomey X.

By

Lemma2.1, there exist acofinalsubset

F

of

D

danet V

of nets with domain

F

suchthat for eachnin

F, vn

s,,and

&v

u. Therefore

v

y. Let

q

pF.

Then

vm

toq,, foreverym F. Since

X

isregul,q y. Since

X

is Hausdo d q p, q x andhenceu x. Since

X

is pseudotopologicM devery universlasubsetof converges to x, As xand hence

X

is topological.

Thefollowingcorollary follows immediately form Theorems 2.6 d 2.8.

COROLLARY.

Let

X

be anoncompact, Hausdo, pseudotopologicalspace, d let X* beitsone-poitconpactification. Then X*is topologicalif madonlyifX* is regul.

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B.J.

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(1988),

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FROLIK,

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Vo1.10(85)

(1960),

168-180.

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SO,

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93-103.

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AARNES,

J. F. and

ANDENAES,

P. R. On nets and filters, Math. Scand. 31

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KENT,

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