On strict and simple type extensions

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VOL. 21 NO. 2 (1998) 239-248

ON STRICT AND SIMPLE

TYPE EXTENSIONS

MOHAN TIKOO

DeparUnentof Mathematics Southeast MissouriStateUniversity CapeGirardeau,Missouri 63701U.S.A.

(Received August 14, 1996 and in revised form November 21, 1996)

ABSTRACT. Let (Y,x)be an extension ofa space (X,x’). pY,let

Of

{WcX:W

x,peW}.

For Ux’,let o(U)={p eY:U

Oyr}.

In1964, Banaschweski introduced thestrictextension

Y,

and the simple extension

Y+

of X (induced by (Y,x)) having base {o(U):U x’}and {U w {p}:p Y,andU

e0},

respectively.The extensions

Y

and

Y+

have been extensivelyused since

then. In this paper, the open filters ./’’ {Wx’:W

_

int_x

clx(U

forsomeU

0},

and

/P

{Wx’:intxclx(W

eOr}

{Wx’:intxclx(W)

,d

’’}

=c{/:/

is an open ultrafilter on

X,

0,

c

’}

onXareused to def’me some newtopologiesonY. Someof thesetopologies producenice extensions of (X,x’). We study some interrelationships of these extensions with

Y,

and

Y+

respectively.

KEY WORDS AND PHRASES: Extension, simple extension, strict extension, H-closed, s-closed,

almostrealcompact,nearcompact.

1980AMS SUBJECTCLASSIFICATION CODES: Primary54D25.

1.INTRODUCTION

AtopologicalspaceYisanextensionofaspace XifX is adensesubspaceofY. If

Y1

and

Y2

aretwo extensionsof aspaceX,then

Y:

is saidtobeprojectively largerthan Y1,written

Y

>

Y

(or

Y

<

Y:

),

provided that thereexists acontinuousmap

f:

Y

-->

Y

such that

fl

x

ix

,the identitymaponX. Two extensions

Y1

and

Y2

ofXarecalled equivalent if

Y

<

Y

and

Y

<

Y.

Weshall identifytwoequivalent

extensions ofX. With thisconvention, the classE(X) ofall theHausdorffextensions ofaHausdorff spaceXis aset.Let (Y,x) E(X)andletpY.If

N,

is theopen neighborhoodfilterofpinY,theset

0’

{N

X:N

Nr

(calledthetraceof

Nr

onX)isanopen filteronX.IfUisopeninX,denote

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In 1964 Banaschewski [1] introduced the extensions

Y#

(resp.

Y+)

thestrictextension (resp. the

simpleextension)ofXinducedbyYsatisfying

Y"

<Y< Y/. Thetopologyx on

Y

(resp.x on

Y+)

has

for an open base the collection {or(U):Uopen in X} (resp., the collection {Uw{p}:pY, and U

0’

).The extensions

Y,

and

Y+

have been studied extensively andhave proved extremely useful regardingsomepropertiesweaker than compactness, suchasnearlycompact, almostrealcompaet,feebly compact, H-closed, s-closed,etc..Inthispaperwe introduce newextensions

]g,

Y",

1/’,

and

’,

study

some of theirproperties, andcompare them with Y,

Y,

and

Y+.

All spaces under consideration are

Hausdorff.

2. THEEXTENSIONS I

/AND I.

Inthis section,we introduce several topologies on Y, and compare themwith

.

Some of these topologiesyieldinterestingextensions of(X,x ’).

DEFINITION2.1.Let (Y,x) beanextensionofaspace (X,’). ForpeYdefine

blp

{W:Wcx’,intX

clxW

0’},

(2.1)

L

p

{W:W e’,W_intxclxUforsomeU (2.2)

LEMMA2.1.

(a)Both b/p and

’P

areopenfilters onXsuchthat

(b) b/P={W:Wex’,intx

clxW

{b/:b/isanopenultrafilteronX,

O;

hi}

PROOF. We prove (b). Let l,U={W:W’,int

xclxW

o/’P}.

If W}[/, then W’and intxclxW_intxclxUforsome U

0’.

Therefore,

intxclx

W

0,

whence W

P.

Thus, W_b/p. To prove the reverse inequality, let Wb/p Then

int

x

clxW

0.

Since int_xcl_xW

_

intxclx(intxclxW) itfollows that intxclxW ,,P. Hence We1U.This provesthefirst

equalityin(b).Thesecondequalityfollowsfrom[9], completingtheproofof the lemma.

REMARK

2.1. Since

0’

0’#=

0’

+[9,10,11],itfollowsthat each oneofY,

Y+

and

Y

yieldthe same

LP

(resp., b/p)forall peY.Moreover,ifZ

E(X)

hasthe sameunderlyingsetasY,and is such that

Y"

<Z<

Y’,

thenYandZinducethe same d’p

(resp., b/p for all p Y. Also,if p q are distinct elements ofYthen

P

and b/p //q.Obviously, ifU

0’,

then int_x

clx(U)

,d’p. Moreover,

U

fJP

if andonlyif intx

clx(U

eb/p

DEFINITION2.2.Let (Y,x) be an extensionof (X,x’).For Gex’,define

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ou(G

G{.p:pY\X,G

eU

p} (2.4)

at(G

{peY:G

e,./}

(2.5)

au(G

{p eY:G

e/.,/’}

(2.6)

TheproofofthePropositiom 2.1,and2.2 isstraightforward.

PROPOSITION2.1.Let (Y,x) bean extension of(X,x’).ThenforallU,V (a)

ot()

t,Ol(X)

Y,

(b)

o,(U)

x

u,

(c)

o(U

:v)

o(U)mo(V)

(d)The family

{ot(G):G

ex’}is an open base for aHausdorfftopology

x

on Yand (Y,

zt)

is an extensionofX.

PROPOSITION2.2.Let (Y,x) be an extensionof(X,z’).Then forall U,V (a)

ou(

=

and

ou(X)=

Y,

(b)

o(U)cx=u,

(c)

o(U

v)=o(U)o(V),

(d) The family

{o,,(G):G

ex’}is an open base for a Hausdorfftopology x

uon

Yand (Y,zu)is an extensionofX.

PROPOSITION2.3.Let(Y,z)be an extensionof (X,x’).ThenforallU,V.’ (a)

a()

f,at(X)

Y,

(b)a,(U):

x

_

u,

(c)

at(U

V)

at(U at(V),

(d)

at(U)

{W:W

ex and

intx

clx(WX)

_

U}

(e)Thefamily

{at(G):G

Ex’}isanopenbaseforacoarserHausdorfftopologyx_o onY, Xisdensein (Y,x or),but(Y,x

t)

maynotbean extensionofX.

PROOF. We prove (d). The rest is straightforward. Let p

eat(U

Then Ued

’’.

Therefore, U_intxclxV forsome V

eO’.

Therefore, there exists Wex such that

peW

and WX=V. It follows that

intx

elx

(W X)

_

U. Conversely, if Wex is such that int_x

clx

(W X)

_

U and p

W,

then WcX

eO.

So, int_x

elx(WcX

.P.

Thisimplies that U ed

’’

and hence p

eat(U

Theproofofthepropositionisnowcomplete.

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(b)

a,(U)cX

int_x

clx(U),

(c)

a,(U

V)

=au(U)ca(V

),

(d)

aN(U

{W:W

ex andWX

_

int_x

clx(U)}

(e)Thefamily

{au(G):G

ex’} is anopen base fora coarserHausdorfftopology x onY, Xisdensein

(Y,x,),but(Y,xo,) maynotbe an extension ofX.

PROOF. We prove (d). The rest is straightforward. Let pea,(U). Then Ue/’ Therefore,

intxclxUeO

.

It follows that there exists Wexsuch that

pew

and WX_intxelxU. Conversely, if Wex is such that

WcX_hatxclx

U and peW, then

WXe0’.

So,

intx

clxU

eO;.

Therefore, U

el

’

and p

ea,(U).

DEFINITION 2.3.Thespaces (Y,x), (Y,x), (Y,x

o),

and(Y,x o)described inpropositions 2.1-2.4 will, henceforth, be denoted by

,

]’,

t,

and

F

respectively. If A

_

Y, then intr,(A) (resp.

clr,

(A) will be denoted by

intl

(A) (resp.,

cll

(A)). Likewise,

in.(A),cl(A),into(A),cl.t(A),into.(A

and

cla.(A

are defined in ananalogousmanner. LEMMA2.2.IfUex’,then

(a)

a(U)

_

o(U)

_

or(U

_

o.(U)

_

o.

(intx

clxU)

a,.(U) a.(intx

clxU

0)

at(U)\X=ot(U)

\X,

anda(U)\X=o(U)\X

(c)

ot(intxclxU)

\X

=ou(U)

\X,and

(d) if Uisregularopen (i.e. U intx

elxU

then

a,(U)

a(U),

andthe equality holdsin(a).

PROOF.Part (a): We show that o(intx

clxU

a.(U),

therestbeingstraightforward. Certainly,

o,(intxclxU)CX=intxclxU

=a.(U)X.

Let peo,(intxclxU)\X. Then

intxclxU

ell’. Therefore, Ue/A

’’,

and p

ea,(U)\X.

Conversely, let p

ea(U)\X.

Then, Ue/’. So,

p

eo(U)

X

_

o,(intx

clxU)

X.Theabove arguments prove (a).

To prove (c), let qeo(intx

clxG)

\X. Then, intx

clxG

e.l whence, Ge/q Therefore,

q

eo(G)

\X.Thus,

o(intxclxG

\X

_

o(G)

\X.Toprovethe reverseinequality,let q

eo(G)

\X.

Then, Ge/q whence

intx

clxG

e.[ Therefore, qeOl(intx

clx

G) X and

o(G)\X_ot(intxclxG)\X.

Hence,

ot(intxclxG)\X=o,(G)\X. The rest of the lemma is

straightforward.

Given aspace (X,x’),thefamily {intx

clxU:U

ex’} forms anopenbase foracoarserHausdorff

topology

x’son

X. The space

X,

=(X,x])

is called the semiregularization ofX. Aspace (X,x’) is

called semiregular if(X,x ’)

X,

THEOREM2.1.IfXissemiregular,and (Y,x) (notnecessarily semiregular) is an extensionofX, then i is an extensionofXsuchthatyt < y.

PROOF.IfXissemiregular, then

o(U)

at(U)

for all Uex’.Hence,

]i

is an extensionofXsuch

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THEOREM2.2.Thespaces

’

and

yu

arehomeomorphie.

PROOF. For all U ex’,

a(intxelxU)=ou(intxelxU)=au(U

implies that xau_xot. Also, if G ex’ and peat(G), then G_intxelx(U for some

UeOf

_//P. Now, if qeau(U), then

intx

elxU

e.m

which implies that Ged

’,

or q

eat(G

). Therefore, p

ea(U)

_

at(G

Hence,

x_a

-

Thisprovesthe theorem.

LEMMA2.3.Let (Y,) be an extensionof(X,x’).Then, forall Gex’ thefollowingare true.

(a)

cl(G)

_

elt(G

cl(intx

elx(G))

(b)

el(G)

elow(G

el(intx

elxG))

(e)

clu(G)=clt(G),

(d) ely(G)

elo(G)

elo(int

x

elx(G))

and

(e)

cl(o(G)) =elo(a(int

x

elx(G))

PROOF. Part (a): Let p

eelo(G),

and let

ot(U)

be a basic open neighborhood of p in

yr.

If peot(U)X,thenpU _intxelxU,P. Therefore,

a(intxelxU

isanopen neighborhoodofp in

].

Consequently,

at(intxclxU)tG().

By Proposition (2.7) (b),

intxelxUtG.

Hence UG ).This inturnimpliesthat

ot(U)G

(),and p

cI(G).

If p

eo(U)

\X,then U

e.g

p. Now,

at(U

is an open neighborhood of p in

t.

Consequently,

at(U)G(.

Therefore,

or(U)

G whence p

el(G).

Part (b): Let p

Clo(G),

and let

o(U)

be a basic open neighborhood of p in

Y.

Since

o(U)_a(U),a(U)

is an open neighborhood ofp in

F

’.

Hence,

a(U)cG.

Therefore,

int_x

clxU

G

,

whence UoG

.

Consequently,

ou(U)cG

:

.

Therefore, p

ecl(G).

Therefore,

clo(G)

_

cI(G).

Conversely,let p

eel(G),

and let

a(U)

be a basicopenneighborhood

ofp in

.

If pea(U)cX=intxclxU, then

o(intxclxU

is an open neighborhoodofp in

’.

Therefore,

au(U)X=intxclxUCG=o(intxclxU)CG. Hence,

p

eclo(G).

Now,

if pa(U)\X, then Ue/ and

o(U)

is an open neighborhood of ofp in

Y.

Therefore,

o(U)cG :

.

Consequently, a,.(U)cG

:

f andp

ecl,(G).

Therefore,

cI(G)

_

clo(G). Hence,

cl(G)

_

cI(G).

The other halfof(b)isstraightforward.

Theproofof(c)isstraightforward.

Part (d): Let p

col,(G),

and let Wbe anopen neighborhood ofpin Y.Then, WcX

₀

_t

shows that

o(WcX)

is anope

n

neighborhoodofpin

.

Therefore,

au(Wc

X)

.

Thisshows that WoGf, whence p

eclr(G).

Conversely, let p

eclr(G),

and let

a(U)

be a basic open neighborhoodofpin

F

.

Then, U

e’’.

So,

or(int

x

clxU

isanopenneighborhoodofpinYsuch that

or(intxclxU)cG

.

Thisimpliesthat

a(U)G

,f.Hence, p

eClou(G).

Therestfollows from (c).

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PROOF. To provethe continuity of the identitymap i:Y \X-->Y\ X, let

o(G)\

X be a basic

open neighborhood of p in Yt\X. Then, Ged

’’.

Hence G_intxclxU for some U

eO

_/’. Therefore,

or(U

) X is anopen neighborhoodofpin suchthat

or(U

X

_

or(G

X. To provethat theidentitymap i:Yt\X yr\_X _is_continuous, _let

ou(G

_X _{be a basic}_open _neighborhood_ofp_in Yr\X. Then ol(intxclxG)\X is an open neighborhood of p in Yt\X such that

ot(int

x

clxG)

\X

ou(G)

\X.Hence,thespaces

Y

\X, and yr\X arehomeomorphic.The rest of the theoremfollowsdirectlyfromLemma2.2.

Let

Z1

and

Z2

bespaces. A map

f:Z

--

Z iscalled0-continuous [3]ifforevery p

eZt

andfor every open neighborhood Voff(p) inZ2, there exists anopen neighborhood Uofp in

Zi

suchthat

f(clz,

U

_

clz2

(V).

fis

ealledperfect

iffis

aclosedmap (notnecessarilycontinuous) such that

f’-(z)

iscompactin

Z

for every zCZ Also,

f

iscalledirreducible

iffis

closedandthereisnoproperclosed

subsetKof

Z1

forwhichf(K) Z Twoextensions

Z,

and

Z2

ofaspace Xare called 0 -equivalent if

there

exists a0 homeomorphism

ffrom

Z

onto

Z2

such that

fl

_x ix,theidentitymaponX.

The nexttheorem depictssomeof the several interrelationshipsbetween thespacesY,

Y,

I;,

Y",

and

t.

THEOREM

2.4.Let (Y,x)bean extensionofaspace(X,x’).Thefollowingstatements aretrue.

(a)The identitymap i:

ya

yisperfect,irreducibleand0-continuous. (b)The identitymapi:Y yu isperfect,irreducible and 0-continuous.

(c)The identitymapi:

Y’Z

--

Y

is0-continuous.

(d)The identitymap i:

Y

--

yt is0 -continuous.

(e)The identitymap i:

Y"

--->yris0-continuous. (f)The identitymapi:Y

--

Y"

is0-continuous.

(g)The identitymap i:

Y"

--->

Y"

is0-continuous. (h)Theidentitymapi:Y

--

Y"

is0-continuous.

(i)The identitymapi:

F’

Y

is0-continuous. (j) The identitymapi:yt y is0-continuous. (k)The identity mapi:

Y

Yis0-continuous. (1)The identitymapi:

Y’--- Ya

is0-continuous.

PROOF. Below, we outlinetheproofs ofsomeparts ofthe theorem. The rest ofthe proofsare

analogous.

Part(a)Since xat T.,i:Y-

Y’

is continuous.

Hence,

i:Y

--

Y’

isirreducibleandperfect.To prove

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STRICT AND SIMPLE TYPE EXTENSIONS 245

clat(a,(int

xclx(VcX))

clr(at(int

xclx(VX))

clr[al(int

x

clx(VX))cX]

clr(intx

clx(V

X))

_

clr(V

Hencei:Y ---)Yis0-continuous.

Part (b): Forall G

x’,au(G

ou(int

x

clxG

x

shows that i:Y ---)

Y

is continuous.Therefore,

i:Y--)Y is irreducible andperfect. Let

o(G)

be a basic open neighborhood ofp in

F’.

Since

ou(G)_a(G),a(G)

is an open neighborhood of p in

I

such that

Clo(a(G))=clu(ou(G)),

establishingthe0-continuityofi:

Y"

--)Y.

Part (c):Toprove the 0-continuityofi:Y ---)

Y#,

_{let p}Y andlet

or(G),G

x’ be a basicopen neighborhoodofpin

Y#.

Then, G

0’

_t/’ implies thata,(intx

clxG

is anopen neighborhoodofpin

F

t

such that

cl(a(int

_x

clxG))

_

clr#(or(G))

Part (d): Let

ot(G

be abasicopenneighborhoodofpin

A.

Then,

or(G

isallopen neighborhood of

Y#

such that

clr#(or(G))

_

clt(o(G)),

establishingthe0 -continuity ofi:

Y"

--)

Yt.

Part (h): Let

o(G)

be a basic open neighborhood ofp in

7".

Then,

o(intxclxG

is an open neighborhoodofpin Asatisfying

cl(ot(int

x

clxG))

_

cl(o(G))

Part (1): Let p

Y

and let

at(G),G

x’ be a basic open neighborhood of p in 7

at.

Then,

G

_

int_x

clxU

for some U

e)’.

So, p

or(U

Now,

elf(or(U))

clr(or(U

X)

clr(U

_

cla(U

_

clo,(at(G)).

Wenow summarizetheresults provedabove in thefollowing theorem.

THEOREM2.5.The spaces Y,

Y#,

IA,

I’,

and

F

at,

arepairwise 0-homeomorphic. Thespaces

A,

and

Y"

are0-equivalent extensionsofXwithhomeomorphicremainders.

ItiswellknownthatspacesYand

Z

are0 -homeomorphic if andonlyif theirsemiregularizations

arehomeomorphic. 11 Hence,wehave the followingcorollary.

COROLLARY 2.1. Let (Y,x) be an extension of a space (X,x). Then, the spaces

Y,,Y,",Ys,

Y,,

and

Y,’

arepairwisehomeomorphic.

Moreover,

Y,,Y/,.

and

y,u

areequivalentextensionsof

x.

3.

THE

EXTENSIONS

YS’,

AND

Y".

Inthis section, we define extensions

A*,

and

F",

analogous to the simple extension

Y+

of (X,x’) inducedbyan extension(Y,x) ofX.The

spaces/*,

7

t, F",

and u* allhave thesameunderlyingset

r’"

as the set Y. An open base for the topology

xz.

on (respectively, x_or. on

F

t*)

is the family x’ {G {p}:peY\

X,

Gedv (respectively, x’o{G {p}:GCd

’’}

).Anopen base for the topology

*

x on (respectively, x on

ya*)

is the family x {G{p} peY\X,G

’’}

(respectively, x’ {Go {p}’G

/’}).

Forany A Y,

cI.(A)

will denote the closureofA in

/’,

with analogous

notations inothercases. Theproofsofthefollowingstatements arestraightforward,and we omit the details.Obviously,thespaces

Y

X,

Y"

X,

Y’"

X, and

Y*" X

are all discrete.

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LEMMA

3.1.Foreach Gz’,

cl.(o(G))

cl(o(G),

and

cl.(ou(G))

clu(%(G)).

THEOREM3.2. Each oneoftheidentitymapsi:

Y+

Y’,

andi:

Y"

---)Y/ is0-continuous. THEOREM3.3.The spaces

Y,

Y*,

and

yu.

are 0- homeomorphic.Moreover,

Y+,

Y’,

and

Y"*

are0 equivalentextensionsofXwithhomeomorphicremainders.

COROLLARY 3.1. If (Y,) is an extension of a space (X,’), then the spaces

Y+,Y/’,’,Y:’,

and

Y.,’"

are homeomorphic in pairs.

Moreover,

the spaces

Y,+,

Y/’,

and

""

are

equivalentextensions of

X

REMARKS 3.1. (a) If P is any property of topological spaces which is preserved under

0-continuous surjections,and if(Y,x) isa P-extensionof(X,x’),then

Y,

Y",

F",

and

Y""

arealso P-extensionsofX.

(b)Theextensions

Y,

Y",

I/’,

and

I’"

introducedabove are,ingeneral, alldistinctfrom

Y,

Y#,

and

Y+.

Itwouldbeinterestingtofind a characterizationofspaces Yforwhich

Y#

Y.

A space Ziscalled H-closedif it isclosed inevery Hausdorffspaceinwhich it isembedded[see 11 formoredetails]. The

Katetov

(respectively,Fomin)extensionofaspace (X,x ’) isthespacev,X (respectively, oX whose

underlyingset is theset X {p:pis afree openultrafilter onX),andwhosetopology has foranopen

base thefamily z’{U{p}:Uep, andper,X\X} (respectively,thefamily

{o(U):U

z’}). The

spaces

r,

andoXareH-closedextensionsofXsuchthat (6X) =r,.X,and (r,X) 6X [3, 6, 11]. In general (oX) ;oX,(rX)

,r,,(oX)’;oX,

and(rX)’*r.X.

Analogous remarks apply to the Banaschewski-Fomin-Shanin extension pX 13]ofaHausdorff spaceX

(c) A spaceZiscalled compactlike,ornearlycompact ifevery regularopencover of

Z

is reducible to a finite subcover. A spaceXhas acompactlike extension if and only if

X

is Tychonoff [14]. Compactlike extensions (=near compactifications) of Hausdorffalmost completely regular spacesX (whence,

X

isTychonoff)have beenconstructedin[2]viaEF-Proximities.ForaHausdorffspaceX whose semiregularization

X

is Tychonoff, a maximal compactlike extension BXofX, satisfying

(BX).

fld(,

isconstructedin[14].If(X,’) isany Hausdorff almost

compl6tely

regular space,andif (Y,) isany nearcompactification of(X,x’),then so are

,

I/’,

and

g".

(d) A space Ziscalled almostreal compact ifevery openultrafilter onZwith countable closed intersectionpropertyinZconvergesinZ[4]. AspaceZisalmost realcompact if and only if

Z

isalmost realcompact [12]. Almost realcompactifications ofaHausdorffspace have been constructed (among others) in [7], and [12]. If (X,’) is any Hausdorff space, and if (Y,z) is any almost realcompactification of(X, ’),then so are

I/,

’, l/’,

and

’.

(e) AHausdorffspace Ziscalled extremallydisconnected ifforeachopensubset Uof

Z, clz(U)

is

open.A space Zisextremallydisconnectedifand only if eachdensesubspaceofZ[respectively, if and only

ifZ]

isextremallydisconnected[see11for moredetails]. AHausdorffspace Ziscalleds-closed if

itisH-closedandextremallydisconnected [8]. AHausdorff spaceZiss-closed ifandonly if

Z

is

(9)

moreover,an extensionYofXiss-closed if andonlyifX is

C’-embedded

inY. If(X,x’) isany extremallydisconnectedHausdorff space,and if(Y,x) isanys-closed extensionof(X,z’),then so are

y,

y*,

and

y

*

REFERENCES

BANASCHEWSKI,

B.,

Extensionsof topological spaces.Canad.Math.Bull. 7(1964),1-22.

[2] CAMMAROTO,F.andNAIMPALLY S., NearCompactifications, Math. Nachr. 146(1990), 133-136.

[3] FOMIN, S.V.,Extensionsof topologicalspaces.Ann. Math.,44(1943),471-480.

[4] FROLIK,Z., Onalmostrealcompactspaces, Bull.DeLcademeie PolonaiseDesSciencesMath., Astr.etPhys./X:49(1961),247-250.

[5] KATETOV,M., Anote on semiregular andnearly regular spaces.Cas.Mat.Fys.72(1947),

97-99.

[6] KATETOV,

M.,

On H-closedextensionsofatopologicalspaces.Cas.Mat.Fys.72(1947), 97-99.

[71

LIU,

C.T.andSTRECKER,G.E.,Concerning almostrealcompactifications, Czech.Math.J. 22 (1977), 181-190.

[8] PORTER,J.R.,Hausdorffs-closed spaces,Q&AinGeneralTopology,Vol. 4(1986/87)

[9] PORTER, J.R.andVOTAW, C.,H-closed extensionsI. General TopologyandAppl.,3(1973), 211-224.

[10] PORTER, J.R.andVOTAW,C.,H-closed extensionsII. Trans. Amer.Math.Soc.,202(1975), 193-209.

[11]

[12]

PORTER, J. R andWOODS, R. G.,Extensions andabsolutesofHausdorff spaces. Springer-Verlag,NewYork,1987.

TIKOO,

M.,

Absolutesof almost realcompactifications, J. AustralMath.Soc.(seriesA), 41(1986),251-267.

[13] TIKOO,M.,The Banaschewski-Fomin-Shanin Extension pX,Topology Proceedings,Vol.10 (1985), 187-206.