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 convergencespaces^arealsodiscussed.

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. Thedomain}and 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 asubnet of s, then

(t, z)

C. If Cisaconvergence structureon

X,

then

(X, C)

^orjust

X

iscalledaconvergence space, and the statementsconvergesto 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.

THEOREM

1.1.

A

convergencespaceXis colnpact if andonlyif everyuniversalnet in

X

converges.

Next,

weintroduce (-subncts and reversibleE-subnetswhichareimportant inourdiscus- 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 a}net 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 of}S such that s ^---,x.

By

Lemma1.2, S_<9 ^s. Therefores isareversibleE-subnet^ofSandhence x lira

inf

^{S. Since}

lira supSC_ lira in

f ^S,

^itfollows froxn Theorem 1.2 that ^{lira sup}S lira in

f

^S.

If

X

is^aset 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

LEMMA

^{1.3. Ifsis a netin}

^X

^and

^A

^{isanonempty subset of}

X,

thens

is.

^{frequently in}

A

ifand only ifsomesubnet ofsis eventuallyinA.

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 a}directed set. Let u

alE,

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

s(iD)

^{fqA and hence}u(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,

^ofaconver-

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. One^{of the}essential 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 h^os

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

,n

D

such that ifp

_>

m, there is q

_>

nsuch that

S

_{v 3} uq ^and hence vv

h-(Sv)

uq.

SO

Thcrcforcv

_<

^u and ^1,c,,,-,;

h-’

^oS

h-’({x,}).

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

- ^X

^{arc cont,bea}conw.rgcnc.^{inuos,X i}SlW^l,’rllic

,.

IfI

K,

^tothen^timsubspacc4is theset of

^X

^ofX.all 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 a}homcomorphisxn 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 map^fi’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 x}inX. 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.

THEOREM 2.4. IfXis Hausdorff and pseudotopological,thenX* ispseudotopological.

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 is}frequentlyin 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 thatalthougla^a convergencespace ispretopological, itsone- 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 for}every ⁿ N. Let be a net withdomain N andlet t,,

s,(n)

^{for each n}

e

N. Then

<

A and t,,

{

^{ifnis odd It}follows 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 n

_ ^D,

^Un

^tn]X.

Since for each n

D,

u,

<

s, andso

un

^--}x. Hence As ^--}x because

X

S.S. SO

is 1)retopological. SinceX* isCOmlact ^{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 hence_q 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.

Spaces

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Vol.23(43) (1988),

135-142.

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FROLIK,

Z. Concerning topological convergence ofsets, Czechoslovak ^Math.

Vo1.10(85) (1960),

168-180.

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SO,

S. S. Special mappings defined on convergence spaces, Journal of the Institute of Mathematicsand

Computer

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Vol.5(1)(1992),

^93-103.

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AARNES,

J. F. and

ANDENAES,

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

(1972),

285-292.

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KENT,

D. C. and RICHARDSON G.D. Compactifications on convergence spaces, InternationalJournalof MathematicsandMathematical Sciences

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PEARSON,

B.J. UnpublishedLectre

Notes,

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