~B~UNIVERSIDAD AUTONOMA METROPOLITANA ~ ~~ . _. _ _ -

Casa abierta al tiempo UNIDAD IZTAPALAPA División de Ciencias Básicas e Ingeniería

DEPARTAMENTO CIE MATEMATICAS Arca de Topología

CONNECTIFYINC3 SOME SPACES

Autores: Tkacenko Michal Tkachuk Vladimir Wilson Richard Alas Ofelia

04.0406.1.0 1 .O I0.W

Coriiiectifying some spaces.

O. T , A LAS , M. G . TIC AC EN I\: O , V. V. Tiíxc. II U I<, R. G .\Vi LS O N

Absfsacf. A IIausdorlF space ,Y is called (countably) coniieclifisble if there exists a coiiiiected

IItiusdorfr space 1' (witli IV\,Yl<<w respectively) siicli that ,Y eiribeús densely into Y. We prove that it

is coiisisteiit witli Z F C that there exista a regular &rise ¡ii itself countable space w h i c h is iiot coiiiitably

coiiiicctiíiable giving t h s a partid niiewer tu ProLleiii 3.9 froiii [ i ] . Oii the utlier Iiaiid we ~ I O W t l i s t

hlartin's axioiri implies that every cuuiitaL1e deiise i i i itselí apace X with rrw(S)<2" is cuuiitably cuii-

iiecliíiabie. We also establish t l iat a separable irietikable spnce witliuut open coiiipact subsets caii be

deiisely eiiibetlded iii a irietric coiitiiiiiuiii.

Keywortú: coiiiiectiliable, countably coiiiiectiliable, coiiiiectifyiiig íainily, reiiiainder, extreindly

discoiiiiected space

C'luss~ficulauir: Priiiiary 54 DOG; Secoirdary 541)25,54C25

O. Iiitroductioii. We stiicly 1h.ii~tlo.ríF spew wliicli atliuit tleiise eiiilxtlcliiigs in c:oiiiic:ctc;rl H:~i i~c l~r i€ spaces. Siicli spnc(is arc c:~llerl c:oiiiiw tifiaLlci tuid tlieir c u i - iiectetl esteiisioiis u e referrd to as coi~ii~.:ctificatioiis. If Y i s a, couitxtiíication of -Y aiid Y \ S is coiiutai>le tlieii Y is called üu w-c.onnectificatioii of -Y u t 1 if S l i u xi u-coiluectificatioii, theii it is called w-coiiiiectifial,le, or coiutably coriiiectifial>le.

Altlioiigli IL cliuncterizatioii of Haiisdoiff coiiiiec tifial~le spaces is still i i i h o w i i , sigiiificruit progiess iii stiitlyiiig coiiiiectificabili ty lias ljeeii achiever1 in [ i] aud [a]. %or exífliiple, tlie followiiig C ~ L S S ~ ~ of Haii:jtlorfF S ~ C C S are sliowii to lie subclasses of tlie class of coiiiiectifiahlc. spxes: - liaracoiiipact, first coiiiitable spaces witli a a-locally finite K - ~ J Z L S ~ (in l iut iculu, tlie iiietric oiies) x i t l !vi tli iio proper open coiiipact subsets (21; - couiitalle spaces witlioiit isolated iioiiits [i]; - Tychoiioff iiowhere locally coiupact spaces wi tli coiiiitalle n-weiglit [ 11;

Also, various ex:uiiples of iioii-~oiuiec t ifiable sp ices are giveii iii tliose pal~ers. It was asked in [l] (Prol>lein 3.9) jvlietlier every coiiiitable clei~se iii itself Hatis-

clorff space is coiiiita1,ly coiiiiectifial>le. We prove that there is a Tyclioiioff coiiii- terexaiiiple i n every iiio(le1 of ZFC for wliicii tliere exist P-poiiits iii / jw\w. Oii tlie utlier 1i:uid we sliow tliat if bí:utiii's u i o i i i lioltls, theii every coiiiitalh cleiise iii itself HaiisclorfF spnce

Of co~use, coiiiitalh coiiiiectificatioiis of coiiiital~le spaces Lave to he iioii-

rcgular. However, we prove tiint every second coiiiitülJle regiilar space wi tlioiit opeii coiiipact siil>spact?.s Iirci a iiietrizable c.oiiiipact coiiiiectificatioii. It was estab- lislietl iii [2] tliüt iiowlieie locally coiiipct secoiid coiiiital>le regiilru spaces llave a Tyclioiioff coiiiiectific. ' a t 1011. '

wi tli 7rw(X) < 2" is coiiiitalAy coii1icxtifi:rLle.

1

1. No tat ioiis aiid teriiiiiiology. All spaces iiiicler coiisitleratioii are assiiiiied to he Haiistlorff. If X is a space t h i T(X) is its topology xitl T * ( S ) = T(S)\{O}. If A c S tlieii T(A,,Y) = {U E T ( S ) : A c U} a i d T(x,S) = T ( { : c } , S ) . Au end of a proof of a stateiiieiit is iiiarlíetl Ly O. If a siibstateiueiit i s proveti, tiieii we wiU us(: tlie sign d. A clopeu siilxet of a topologiczd space is cdlecl proper, if iieitlier it iior its coiiipleiiieiit are empty. Tlie letter A is iisetl to cleiiote the clingoiial prodiict of iiiappiiigs. If S is a space zurl S c :i is couiitahle, theii S -t -c says that the sequeiice S coiiverges to X . We use tlie nlhreviatioii 13L for Bootli's Ieiiiiiia. A space X is cdecl Urysolui space if x, y E S&z # y iiiiplies tlie esisteiice of U, E T ( z , ..Y) ;uid U, E T(y,X) with Ü, n Ü, = fl. If 'r is a cartliiial, tlicm expr is 2'. All otlier iiotatioiis are sttuiclucl.

2. Coimectifyiiig couiitable spaces. A coiiiitalJle cleiise iii itself space is coiiiiectifiable ant1 i t is w-c.oiiiiectiiinLlti if i ts 7r-wciglit is couiitd~ie. Tliese results were oltaiiiecl iii [I], wliere the foliuwing questioii was posed: [l, Prubleiii 3.91. Is every coiiiitülile space wi thout isolated poiiits couiitzhly coiiiiectifialile? We show that a counterexample exists if /Jw\w lias P-poiiits.

2.1. Propositioii. Let S Le n coiiiitable Tydioiioffspíice witlioiit isolated pohits. Siippose that X is tleiise iii a coiiiiectecí ::p:ice Y. Let

Then 3 y consists of iioii-eiupty coiupact subsets of /JX\X rrud (1) for every proper cdopeii O c PX we have F n O # (b # F n (/JX\O) for soiiie F E 3 y ; (2) if X is extreiiidly tiiscoiuiectecí, the11 F y is a tiisjoiiit fuuily.

Proof. That every inember of 3 y is coiupact aiitl iioii-eiiipty is evicleiit. Tlie space Y Haiiscloiff so tliüt for aiiy 2 E ,Y <wcl y E Y\X there is ;L U E T ( y , Y ) with c I y ( U ) 3 :C aid lieiice c l ~ ( U r l X) ;?I X . Therefore c l p , ~ ( U n X) 3 :C anti the set F,, = n{cl,j.y(U n ,Y) : U E T ( y , Y ) } tlocs i i d coiituiii ;t. This sliows that ali elemeiits of F y lie outside A'.

If O is r? proper clopeii siilxet of /J.X then so is O n A- (iii A-). Suppose that for all y E Y\X we have Fy n O = fl or Fy n (/JX\O) = fl. Tlieii for every y E Y\X tlicie i s a U, E T(y,Y) with (X n U,) n O = (d or (S n U,) c O tlie set Fg beiiig ai iutersectioii of coiiipact siiljsets of / J S .

The s lmx Y is wiiiiecteC1, so there is :L poiiit y E c l y ( 0 n S ) n c.ly(X\O). Ilciice U, n ( O n ,Y) # fl :nit1 U, n (S\O) # 8 wliicli is a coiitratlictioii, proving (I).

If X is extreuidly tliscoiuiectetl, tlieii for distiiict y, z E Y \ X take U E. T ( y , Y) and V E T ( t , Y ) with U n V = (d. Tliell c lp ,~(U n S) n clp,y(V n X) = rd Ly tlie extreiiial discoiiiiectetliiess of X'. Tlierefore F, í l Fz = @ atid 2) is proveti. 0 2.2. Corollary. If X is a coiiiitable Tyt~ioiioff'coiiiit¿ibly coiiiiectifiable space, tiieii SS\X hc?s a tleiise a-coiiipact sribspace.

2

Proof. Let I' iii 2.1 is coiiiitaLle zuicl

a coiiiitalde coiiiiectificatioii of S. Tlieii tlie frriuily 3 j f clefiriecl

UFy C / J S \ X C da , . (U3i . - )

by (1). cl 2.3. Corollary. Let be a iiiodeí of Z F C in rvliicdi there is a P-point iii u* = @u\,. Theii tliere is n coiiiitnble cieiise iii itself space -Y E iL1 ivbicíi is not couiitabíy coiuiectifialle.

Proof. Let .Y = C,, where C, is tlie space, coustriictecl iii [3] iisiiig P-poiiits. Tlie Reiiiuk 1 of [3] states that /-IS\S lms 110 cleiise cr-conipact siibset. Now 2.2 sliows tiiüt x can iiot I X coiiiitnIJiy coiiiiectifiaI:~e. O

We wish to tliruik .J.Porter for liriiigiiig tliis exuiiple to our attention. Oiir iiest s t q ) is to cliow tliat tliere ~ U Y siifficieiitly iiiiuiy cx~iiiitalh couiitaljly

coiiiiectifialh spírces.

2.4. Proposition. Let X be a TycJioiioLf couiitnLíc rimse iii itsefl space. Suppose that X h a s a Coiiipactih'catioii b S for rviiidi there esist coiiipíict sets F,,, 11 E w with the foliowizig properties: ( 1 ) F,, n F,,, = ib for tlifieseiit 7 n aid t i ;

(2) F,, c bX\S for all i t E u; (3) for every p,Ul. (U, V) of'uoii-empty cíisjoiiit opeJ1 siilsets of b S with U U V dense iii 6 S there is ai i i E w with F,, n U # l4 f F,, ri V. T h i X is coiiiitnbly coniiectifiable (aid we will cdl tlie f r u d y 3 = {F,, : 18 E u} coiuiectifyiiig for X).

Proof. Let Y = X U {y,, : n E u}, yIi 4 X where X is o p a iii Y a i d a base of opeu iieigliboiirlioocis of a poiiit ytr coiisists of sets U , , ( W ) = {IJ~,} U ( WnX), wliere w E S(F,,, U).

We i ieed oiily prove that Y i s HaiiscloríF aiid coiiiiectecl (the cleiisity of X iii Y being clear).

To selm-ate y,, ii-oiii ail x E X take auy W E T(F,, ,bX) aid U E T(x,bX) with U n W = í4. Tlieii O,,(T.V) n (U n ,Y) := O. If T I L # TL tlien tdce tlisjoiiit IVl aid W, siich that T.v, E T(F,,, b-Y), W2 E T(F,,,, OX). Tlieii O,,(TV,) n O,,(W,) = O so Y is Haiisclorff.

To clieck tliat Ir i s cuiiiiectetl, take m y proper clopen U c Y. The set 01 = O n X is clopen iii A-, so tliat t l iere are disjoint opeii sets U ailel V i i i blY siicli that U n X = Oi aiid V n S = -Y\Oi. It is ~lei~tr tllat cl,,,~(U U V) = bA- SO ~ t : c:aii IS^

(3) to &it1 an t t E w with F,, n U # ib # F,, f-i V. It follows froiil cl,,,y(Ol) = clb,~(U) c1bX(v) = &s(X\01) that F,, n clb,y(O~) # fl # F,, n cl,,s(~~\O1). Heiice

VV n Oi # O # TV n ( S \ O i ) for :uiy tV E T(F,,, OX). This O,,(T.v) n O1 # O =ti

O,,(T/V)n(X\Oi) # fl for dl bV E T(F, , ,bX) so y,, E ~ . l ~ ~ ( O ~ ) n c i ~ ( ~ ~ \ ~ l ) wliicli is a coiitradictioli. U

3

2.5. Corollary. Let S be a coiiiitable Tydioiioff space rvitlioiit isolated points. Siippose that S íins a coiiipactificatioii b S for wliidi tliere exists a sequence { ( x , ~ , yll ) : 71 E u} with the followiiig prolxrties: (1.) { x , ~ , y , : } c L Y \ X for ail 71 E w ; (2) {x , : , y I l } n { x , , ~ , y I I l } = ib for tlifiereiit r n aid 71;

(3) for every pair ( U , V ) of iioii-empty ope~i siibsets of ¿JX there is ~ I J I 71 E w with X , & E u riiitl 7J11 E v. TIiai ,Y is coiiiitnbly coiiiiectifialle (aid we will caí1 tlie seqiieiice {(x,~, y,,) : 71 E w} stroigíy coiiriectifyiiig for S).

Proof. Let F,, = {xI l , y I l } a~ici apIAy Propositioii 2.4. O 2.G. Corollary. Let S be a coiiiitalle dense hi itself estrenidly tliscoriiiected reg- i i l a s space. Tlieii S is coiiiitnbly coiuiectifiable iff t1iei.e is a fluiiily {F,, : 71 E w }

\vi tli the fbiloiviiig prcqwi-ties: ( I ) F,, is í~ cwiiipact subset of /jX\S for di ia E u; (2) F,, n F,,, = (b fbr cfifkreiit 711 aiid it; (3) i f U is a pi-opa d o p i slibstit of /jX, t h i F,, n U # 8 # F,, n (/)-‘.‘\U) for some 71 E u.

Proof. If ..Y cnii Ije cleiisely euil~eclclecl into a coiiutalAe coiiiiecterl Y, tlieii tlie faiiiily 3 y coiistriictetl iii 2.1 has I>roperties (1)-(3) if we eiiiiiiierate it with w. Tliis prc)ves iiecessity. If {F,t : 11 E u} satisfies ( 1)-(3), tlicii let l S = /),Y ruitl apply 2.4. 0

The followiiig re”sisiút lias lieeii receiitly 1uiiio11iici~c1 by .J.H.Porter. We are iiot aware of tlie iiietliotls iised iii l is proof, hiit we give m e here to illustrate tlie iisefuliiesa of 2.5.

2.7. Corollary. (J.H.Portei-). Given a i ii.ifiuite ordilid /j ,< 2“, let 114- be a secorid coiiiitnlie r-egiiiai. space with Ih.rcll > 1 fol. iill Q < /j. If S is í i cmuritabie dense subset of J-J{Mn : cy < /3}, then i t is w-coiiiiectifiable.

Proof. If /j is coiiiitülhq tiieu X lias a coiuitahle weight a ~ i d we cui apply the relevait resul ts of [l].

If ,/3 > w, tlien we iiirry assuiiie that all Me’s are coiiipact, iiietrizallle aid tleiise iii tlieiiiselves for if 110 t we c;ui rq11a.c~: kin Iiy ;I iiietrizalde coiiii>~~~tificatioii of Me tuid tlieii coiisitler tlie p r c d ~ i c : t of dl tlisjoiiit coiin t&iy iiifiiiitc: siilq~rodiic ts

We are goiiig coiistruct ír sequence { (x ,,, y t l ) : 7~ E w} ~ L S iii Corollary 2.5, wiiere U = M . Let xu : Ad -+ M u Le tlie iiatiirai projection. Usiiig t l ie coiiiitaLility of xu(X.) find for every x,, E X = {xi : i E w } í~ coiiiitüljle iiiiiiiber of sequences S;:, = ( t : i Ik : k E u} c M o \ ~ u ( x ’ ) such that. (i) S,’,’l -+ rO(x,,) for every r t , m E w; (ii) i;&, # t;:lq i f p # Q zuitl S,’,!l, n S,’:12 = II :if t i t1 # 7112;

(iii) for all m], 7 1 ~ 2 E w we Iiave S,’:;, n S;::, = 0 if

of Ad = fl{ Ad” : cv < / j } . I

# 712.

4

Now for every pair ( 7 1 1 , n) E (w x u ) tlefiiie íi seqiieiice T: c i\l\S as follows: T:' = { s : : , ~ : k E u } wliere S ~ ~ , ~ ( ~ Y ) = xll(cy) for a l l a > O uitl S : : ~ ~ ( O ) = t : : lk . It is cleaw tliat Tl',: -+ .c,, for every 711 E u. EPor every pair ( I I Z , I L ) E w siicli that 712 f tt let F(tit,tt) = { { . ~ : ~ ~ , . q : : , ~ } : k E u}. The set F = U(F(711, I L ) : I n # t ~ ; 7 1 2 , 71 E u} is couiitable, so F = {{u,,, u,,} : 7~ E k~}. I t is easy to check tliat the extension 6 X = Ad mcl tlie set of pairs { { u,,, u,'} : t~ E u} satisfy ail tlie coiiditioiis of the Corollary 2.5 for s. o 2.8. Coroliary. Any coirii table Tyciioiioff spac:e c a i Le embedried (maybe Iiot deiisely) into a coiiiitnble c-oiinected space.

Proof. Iiirieecl, cuiy siicli space eiiilxds into D = {O, I ) ' ~ . ~ c í c i to i t soiiie coiiiitalile tleiise subset of D a i r i iise 2.7.U

2.1). Tlieoreiii. [BL]. Every coiriitírlle (le~ict! iii itself (Haiis(foríf) space .Y rvitli m o ( S ) < 2" is w-coiuier:tifiable.

Proof. \Ve first prove tlie tiieoreiii for regidru. spaces. To tliat eutl let us reduce i t to tlie case wheii w ( S ) < 2". This will Le achieved with tlie followiiig

2.10. Leiiiiiin. Let (S, t ) Le a coiintabíe regirl,arspace with í i x-Laie 7. Thai t l iere exists n I-egiiiar TI-topology t* oii X siitdi that

Proof of the leiiiiiia. Being coiintalde X is lieretlituily Lindelof. Hence for every opeii set V in X tliere exists a coiitiiiiioiis fiiiictioii fv hoiii X to the w i t segiueiit I = [O, i] such that X\V = f ~ ' ( 0 ) . Let f be tlie tliagoiid prociiict of functioiis f ~ , V E 7 U p, wliere p is a coiintable fruiiily of opeii sets iii X sepíuxtiiig poiuts of X. Let 3' = -y U p . Tlieii f i s a, coiitj.iiiious one-to-one iuappiiig of X iuto Id, ruicl 13') < 171 . u. Let f3 Le a, base of I6 with IO) < Idl. Tlieii tile topology t* on X geiierated by tlie lmse { f-' (O) : O E U} iias tlie required properties. d

2.11. Leiiiiiia. (BL]. Sizppse that S is a coirntnlJle regda- space withotit isolated points, a i t f w(X) < 3". Then tlie immilitfer / jX\S is sepiuabíe.

Proof of the leiiiiiia. By a tlieureiii of Eda, Kaiiio u t 1 Nogiira [4], iinder Booth's leiiuiia every coiiutalde regiilar iioii-scattctred space Y with w ( Y ) < ZW coiitaius a copy of the ratioiials Q. Le t 7 be a iiiaxiiiial disjoint fííiiiily of siibspaces of X lioriieoiiioilhic to Q. Tlieii 2 = U7 is tleiise iii X. We claini tliat for each z E 2 oiie (:ai fiiicl a seqiwiice S, c dX\X coiivergíiig to 1. Iiicleerl, clioose C E 7 with z E C. Tlie poiiit I 1ia.s coiiiital>le c:liarac;:er iii I< = cl~,yC, so tliat tliere exists a seqiieiice S, C I<\X coiiverging to t. Now put D = ü{S= : I E Z}. Tlien D is coiiiitable aud dense in ax. A 2.12. Leiiiiiia. Let S be a coimtabíe regiilar space ivliicdi is clase in a regirlar space 2 mid sirdi that Z\X is separable mid also dense iii 2. Theii X h<as a stroligly conuectifying seqiieiice ( a i d is tlierefore w -coiii~ec tifiable).

c t* c t ímd w ( X , t ' ) < 171 w.

5

Proof of t h e leiiiiiia. Pick a coiiiitalile D cleiise iii Z\X. Botli D ruicl X are cleiise iii 2. For eacli ;c E S we liave x(:c, 2) =: x(x .X) < w ( S ) < 2". Siiice D is tleiise aid coiiiitable, BL iuiplies tliat for endi L E X tliere exists a seqiieiice S, c D, coiivergiiig to :c [SI.

Let {x,, : I L E w} lie a.11~ eiiiiiiierrrtioii of S. There exists ai oiito iiiappiiig y : w\{o} -t w x w siicli tlint y ? - ' ( k , L) is iiifiuite for each pair ( k , 1) E w x w. Let Fu lie a i y siilset of Z\S with IF01 = 2. Siippose that for soiiie 71 > O we have rilready clefiiietl clisjoiut siilisets F,,, of .Z\X for all in < t i sucli tliat IF,,,l = 2. If 4 7 1 ) = ( k , I), clioose poiiits y,, E Szk\ U,,,<,, F,,, u t 1 z,, E Sz,\ U,,,<,, F,,, with

Tlie iiidiictive coiis triic tioii Ijeiiig accoiuplislietl let 11s verify tliat tlie faiuily F = {F,, : ti E w } is stroiigly coiiuectifyiiig for A-. For U, V E T ( / j X ) pick x/i E S n U aid LcI E S n V. By tlie dioice of 9 tlit? set Ad = y - ' ( k , I ) is iiifiiiite u t 1

ru'e fiiiite wiiile tlie iiifiiiite f;Liiiily {F,, : I L E M} is disjoiiit; Leiice tliere exists ai

Now let us finish the Tyclioiioíf case of Tlieoreiu 2.9. Deiiote Ly t the topology of X aid choose a. ~-1ií~stf 7 for X with 171 <: 2". Apply Leiiiiiia 2.10 to find a regula Sl-topology t* fi>r X such that y c t* c t aid I t * \ < 171 < 2". Let Y = ( X , t * ) :uid denote Iiy id tlie itleiitity niqipiiig of S oiito Y. Let f : /jX + /)Y he tlie coiitiiiiioiis exteiisioii of id. By Leiiuiia 2.12 tliere exists a stroiigly coiiiiectifyiiig faiiiily F y for Y with üFy c /W\Y. Now let 3,y = { f - ' ( F ) : F E Fy}. \Ve claiiii that F,y is coiiiiectifyiiig for X.

Indeed, let U lie a proper clopen siil>:jet of / I S aid V = [jX\U. There exist U1, VI E 7 siicli tliat U1 c U aiid Vi C V. 13y tlie clefiiiitioii of tlie topology t* o11 Y the sets f(U1) aud f ( V , ) arc opeii iii Y, lieixe tlieir closiu-es iii /)Y liitve iioii-eiiipty iiiterioi-s. Coiiseqiieiitly, tlieie is ai F E 3-1. wliicli intersects Iiotli tliese interiors. Therefore f-'(F) n U # (b # f-'(F) fi V ~ ~ i i c l so F d y is a coiiiiectifyiiig fííuily for A-. A

Finally let u s tuni to tlie geiieral case. Oiir iiiaiii weqioii will Le the followiiig leiiiiiia, wlicli seeiiis to lie iiiterestiiig in itself.

Y I I # Z?, and pi1t F,, = { yn, 4 1 } .

F,, n s,, f (3 # F,t n S,, for P ~ I 12 E AL I ? ~ ~ ~ ~ ~ ~ ~ ~ ~ x ~ ~ ~ I ~ ~ , ljc.)tli sets S,, \U U C ~ S,,\V

7~ E n.r S ~ I C ~ ~ t h t F,, n U # tl # F,, n V. 1 1

2.13. Leiiiiiia. [DL] Let X be a coimtablfe dense iu i tse-space with w(X) < 2". Theu X has a c h s e regular siibqxice.

Proof. Let 13 he EL base iii S with 1L31 < 2". Tlie family C of lioiiiitluies of eleiiieiits of U lias tlie power less tliaii coiitiiiiiiiiii ruicl for every fiiiite siibfaiidy 7 of C we liave 113\ U 71 2 w for every 13 E U. Tliis eiialiles its to use the Bootli's leiiima to h c l ;L siiliset Y c X siic4i tlint Y n 13 is infinite for rill B E U aiicl Y n C is finite for ali C E C. The first cuiitlitioii iiiiplier; that Y is dense iu A', wliile the secoiicl says that Y 1ias a Lase, ail eleiiieiits of which have fiiiite 1ioiiiiclai.ies. This implies regiilaiity of Y. Iiicletxl, if y E 'I' ; ~ i i c 1 U E T ( y , Y ) , tlleii y E bV c U for soiiie VV with finite Ixxiiiclary. Let V E ??(y, 1') sepa.ri.ate y froiii this boiiiiclary. Siicli a set

G

V exists since Haii~clorffi ie~~ Y is Haiisdorff. It is clear tliat d y ( V n W ) C U aiid we are rloiie. d

flow if -Y is Haiisclorff tuid m u ( X ) = T < Z", trde. soiirc: ir-Lase y iii S with IyJ = T niid fiiitl sc~iiitl couiitnlde fiuiiily rj of ope11 siilm!ts cjf S sepuatiug rtll p i r s of poiiits of S. GcJiier;Lt(: ;I tol>olugy ?; 0 1 1 S by tlie fii~iiily -/ U 6. Tlieii Ti is fI¿~ii~~loriF, w ( S , Ti 1 = T i d y C TI. By Leiiinia 2.13 tliere is ri cieiise i.egulu

Forevery point :c E S\Y let Fz = { U n Y : U E T ( . c , S ) } aiid F' = n{clfiyA : A E Fz}. It is clear tliat t l ie set 'H = U{Fz : .L E -\-\I.-} c $Y\Y. Tlie space 2 = [jY\H is i'ecli r:oiiiplete uid Y c 2.

We clrriiii tlrnt Z\Y is sq)uable. Iiideed, if P c Y is a c:opy of ratioiials, tlieii p1 = c l z ~ is &cii coiiiplete, so tliat PI \P i s tleiise iii P I . ow apply the fact tliat every poiiit of P 11;~s e-oiiiitnlJle cli;iracter iii PI to coidiirle tlint P1 \P is sepual)le ( w e tlie 1)ruof of 2.11). Pick a couiitai>lc clisjoiiit faiiiily /L coiisistiiig of copies of ratioiials lyiiig iii Y- with U I L deiistf iii Y a i d for w c ~ y P E p clioose a coiiiitable A p c clzP\P tleiise iii clzP. Tlieii tlie set A = ü(Ap : P E p } c Z\Y is cleiise iu Z\Y.

Now use leiiuiia 2.12 to hit1 a couiit,al>le stroiigiy coiiiiectifyiiig family for Y coiisistiiig of two-eleiiiriit siilxets of Z\Y. Let U,, I N tiit: faiiiily of correspoiicliiig opeii filters 0 1 1 Y. or eacil U E U,, fiiicl U E T ( S ) wit11 U: n Y = U aiici put Y,, = {U : U E U,,}. Now let 2 = S ü {u,, : n E u} wliere n 1,ase at p,, is { Z h I } u u : u E L}.

To prove that 2 is tlie reqiiiretl coiiiiectific~ltioii, ol~seive first tliat U n V = B a U n V = ib wliicli c.leuly iiiiplíes ihat in -Y U {pli : 7~ E u} aiiy two points p,, # p,,, caul l x sqxu-arterl. That niiy two poiiits of S c:íiii he selxiratetl is evitleiit rtiicl tliere c a i lie 110 prolhiii in selxuatiiig pciiits of Z \ X and Y. Filially, take .wy :c E X\Y aut1 ? L E u.

Let P,, he tlie two-point set iu Z\Y, geiieratiiig the f i l ter U,,. It follows from PI, n F, = 8 tliat there is an opeii iieigiibourliooc1 U of the p i n t x iii X' a i d V E T(P,,, /W) witli (U n Y ) n (V n Y ) = 11. Let vi I J ~ a iieiglilmiirliootl of plr with VI n Y = V. It is clear tlint U aid V, &re disjoint iieígli1,oiiriiootis of z aid p,, respectively aid we trstaLiislied lierewi tli tlie Haustlorffiiess of 2.

Filially, if U is a proper clopen siil~set of S U {pl, : 71 E u,}, tlim fiiicl V, W E 7 ~11~11 tliat V c U :ut1 IV c S\U. Tlie sets V n Y ~l TV n Y : ~ e iioiieiiipty ope11 clisjoiiit siillsets of Y. Heiice tliere is íui 78 E w witli every e1eiiic:iit of U,, iiitersectiiig h t l i V aiid W . Therefore pI, E clz(U n -Y) n c l ~ ( X \ U ) aid we proved tliat Z is coiiiiected tliiisfiilisliiiig Tlieorem 2.9.

siii,space 3,- of (S, T, ).

-

- -

2.14. Reiiiark. It i s iiot yossibie to tlrop Booth's Ie~iiiiia fioiii the hypothesis of Theorem 2.9. IiitleetI, there rue inodeis of Z FC with P-poiiits iii j;lw\w of character less thai 2" (I/ (we t h i k s. bktsou for briugirlg this f k t to 0111- atteiitioii). Now k t X = G", ivl1ei.e Gw is the space, <*oiisti*ii<:tecl iii 131 using P-pii i ts . I t is iiiiiiietli¿rte

'

í

from the definitioii of C,, that the weight of X i s less than continuum. Use Corollary 2.3 to see that S is not u-coiuiectifiabíe..

The following result extends what is howxi for countable spaces (see [i]) to some uncountable ones.

2.15. Propocitioii. Let X be a dense in itselfHausdodTUrysoh space math 1x1 < 2". Then x is (not u e c e s s d y countably) comectifiable.

Proof. Theoreiii 2.2 of [2] states that a space X is connectifiable in case no proper clopen subset of X is feebly compact (:!there is no infinite l o c d y finite families of xionenipty open subsets of X) and the uiuuber of clopen subspaces of X is less than or equal to exp(exp(u)). Of course, tlie niiniber of all subspaces of X does not exceed exp(exp(u)), so we llave to prove only that no proper clopen subset of X is feebly coinpact. Indeed, if some proper clopen U c X were feebly compact we could use the standard procedure to coiistruct a Cantor tree of regular closed sets in U (i.e. take two closiue-disjoint open subsets of U and the same inside ea& of them and so on). Feeble compactness clearly implies that every brancli of tliis tree has non-empty intersection a i d so IVl 2 2" v r L & is a contradictioii with 1x1 < 2".0

2.16. Corollary. Let X be a dense iii itself Tyúionoff space m'th 1x1 < 2". Thm X is connectifiable.

2.17. Remark. The proposition 2.15 can not be proved for Hausdorffspaces (;.e. the word 7 7 U ~ ~ ~ h 1 ' can iiot be hopped from the hypothesis) because there even exist H-closed dense in themselves Hausdodspaces X of power less than continuum. For any s u d X tlie space X @ X is not connectifiable.

3. Trying to construct Tychonoff connectifications. All spaces consid- ered in this section will be Tychonoff. In tlie paper of J.R.Porter and R.G. Woods [2, Theorem 5.71 it is proved that any nowlieire locally compact separable metric space has a Tyclionoff coiinectificatioii. Tlie following result strengthens this theorem.

3.1. Theorern. Let X be a second countable Tyúzoiioff space without non-empty open compact subsets. Then there is a ziletrízable connected compact Y with X c Y = x. Proof. This will follow from several lemias .

3.2. Lemma. Let X be a secoiid countable non-compact space. ?'lien there is a inetrizable compact Z sud i that X c Z =

Proof of the lemma. Tlie space pX\X Iias no isolated points for otherwise some closed neighbourhood of a y E pX\X would consist of a secoiid countable subset plus {y} which would imply existence of a countable network in this ueighbourhood aid hence its metrizability. But then z,, +e y for some sequence S = {zn : n E u} C

and Z\X has no isolated points.

8

X wliiltt tlie closiirti of S iii / 9 S is lioiiieoiiioipliic to /jw tlie sp;~ce A- heiiig iioriiiad. Tliis gives n coiitr;ulictioii proviiig tliat tliere : i I e iio isolatcxi p i i i t s iii /jS\X.

Let 2" be m y i i i e t r i z a l h coiiiP;ictificntioii of S. Let -40 I J ~ tlie set of isolated pii i ts of Zu (wliicli of co~irse Lias to Ix couiitsl~le) uicl iru : /)-Y --t 2" the iiütiird iiiap. For :my 2 E A0 pick tliffereiit t , , u - E TO' ( 2 ) niicl a coiitiiiiioiis map f: : /)-Y 3 (O, 11 witli f 2 ( t Z ) = 1, f=(u=) = O. Xow let

7r; s ; ; - , x l l i. 1 7T' 4 Z " t o Z p - - Z p - - . * . -z,, z...

of iiietrizaljle coiiipnctificatious of ,Y siicli tliat ;7i" t.1- = i(1-y. Now we liave

We claiiii tliat tliat t l ie space 2 = l h Z,, is what reqiiirecl. bicleetl, i t is evicleiit,

that Z i s ai1 exteiisioii of S. If z E Z \ S is isoiated, tlieii tliere i s íui 11 E w with cp,,(z) isolated iii Z,,\S (liere: (p,, : Z t Z,, is i;t-tli liiiiit projection) Ijecaiise v,,(Z\.Y) c Z,,\X for ¿dl 71 . Now I+-J,,(L) is iiot oiily isolated iii Z,,\S h i t ] ~ ; ' ( I + J , , ( Z ) ) ~ = 1 wliicii iiiiplies ~ ( i r ~ ~ + l ) - l ( ~ , , ( z ) ) ~ = 1 wliiie tliis coiitraclicts (*).A 3.3. Leiiiiiia. Lct 2 Le :I second coiiutrrbfe space with iz totdíy Loiindeti metric p. Let 7 = {U C 2 : U # (d # Z\U aid U is clopen iii Z aid />(U, Z\U) > O}. Then fix every E > O the scit {U E y : /)(U, Z\U) > E} is fiilite.

Proof of tlie leiiiiiia. Iiicltied, if it were not so, tlieii tliere woiiltl liave beeq u iiifinite y' c y with />(U, Z\U) > E for {ill U E y'. Lvt 21,. . . , z,, I J ~ an :-net iu 2 with respect to tlie iiietric /I; t l is exists by tlie total I,oiiiicler.Iiicw of /J. Now for every U E 7' if r; E U tlieii

e

But there are oiily fiiiitely iiimy siiljsets of tlie fiiiite family { O,(z;) : i = 1, . . . , I ¿ }

so tliere will he two tliffereiit U, V E 7' with

u = U{Oc(2-;) : i E A} = v for some A c { 1, . . . , I ¿ } , wliicli is ;I, coiitrirtlictioii. A

9

Proof of tlie leiiiiiia. We llave a closed rlecoiupositiou. F of 2 coiisistiiig of one- or two-poiiit siilxets of 2. To prove tliat the relevant qiiotieut iiiap gives a Haii~clorff space (that's all we crctiidly iieerl) we iiiwt clieck that for aiiy F E 3 aiel for m y opeii U 2 F tlieie is LL V E T ( F , 2) siicli tliat G n If # fl iiiiplies G' c U for aiiy G' E J: (see. [G, page 931).

Clioose tul E > O with O,(F) C U. Let V = 0,12(F)\A, wliere A = {G E 3 : tliruii(G) >, t}. Tlic set V is ai opeii iieig;liLoiirliooc¡ of F tlie set A Ixiiig finite by liiii p ( : t l 1 , y,,) = O. It i s straiglitforwaxd chat V is üs reqriirecl. Ll

',-+a3

Now let iis take i i p to tlit? proof of tlie Tlicoreiii 3.1. Using leiiiinü 3.2 hid a iiietrizuble (with n iiietric p ) coiiipact 2 2 S with X

deiise iii 2 ruirl Z\S perfect. Tlie set y of ;.dl proper cilopen siillsets of 2 i s coiiiitable aid p(U,Z\V) > O for cuiy U E y. Let 11 = { U n ( Z \ S ) : U E y}. All eleiiieiits of 71 are 1>roper, I>ecaiise tliere are iio open coiiipact subsets of S. Clearly, for aiiy U E 71 tliere is ax1 > O siicli tliut />(U, (Z\X)\U) > E ~ J . Hence Iiy leiiiiua 3.3 we have liiii p(U, (Z\S)\U) = O.

U€') Let 71 = {Ull : 72 E u} niicl let :G,, C: U,,\X, y?, E (Z\S)\UIl Le siicii that

P ( : h , % I ) < 2p(U,,,(Z\X)\UI1) for all n E u. Using peifectness of Z\X we c m choose {x,,, y?,} c Z\A- iii siid1 a way tliat {x,,' yll } n {xl,,, g,,,} = (4 if 7 7 ~ # 11.

Now itleiitify xIt ruic1 for all n E u. The resultiiig space Y = q(2) will Le a coiupact iuetriza1,le (Ly leiiiiiia 3.4) exteiisioii of S.

We c~aiui that Y is coiinecticl. Iiicleecl, if U is a proper ciopeii subset of Y, tlieii qr- ' (U) is a proper clopen s d m t of .Z :uid therefore q-'(U) n (Z\X) = U,, for some 7~ E w wliicli is iiiipossiLIe, Iie~a~ise. q-'(U) is satiirateci with respect to { {x l l , y,,} : 7~ E w}. This coiitraclic:tioii proves tlieoreiii 3.1. 0 3.5. Corollary. Let S be a Iocdly separable metric space witiioiit open compact stiLspaces. Theii A- Las íi TycJioiioff' coiiuectificatioii. .

Proof. It is well k i i o ~ i i , tlint every siich space is a discrete. iiiiioii of separ.al>le iiietrizalile spaces. Evitleiitly, iioiie of tliese c:lopc?ii sepiual~le iiietric suiiiiuaiitls lias l>rcqit-'r c o i i i p ~ t ol~eii siihl>aces. Heiice we ~ í u i iise thoreiii 3.1 to tlciiisely eiiiljetl X into a discrete iiiiioii of coiiiiectetl iiietrizal>le coupact space5. Now every siimiuaiitl of tliis iiiiicw has :L poiiit which is iiot iii X. Pick i t aiid itleiitify dl tlie poiiits thus cíioseii. Tlie resiil tiiig space will lie a coiiiiectecl Tyclioiioff exteiisioii of X.U

4. Foriiiulatiiig uiisoived probleius. Li tliis section we pose twenty iiaturd qiiestioiis we tlicl iiot siicceetl iii solving w1iil.e workiiig at t l is paper. Of course we

10

J

4.1. Questioii. Is it c:oiisistt?iit with ZFC' that w(qv c ( i r i i i tn í . de tleiise iii itself Tydioiioff' sl>íi(:e is cotiii tably coiiiiec tifiable ?

4.2. Questioii. Is it coiisisteiit with Z F C that every zoiiiitalle tfeiise iii itself Hiitisdoiff Uiysohii spa(:e is coiiiitably coiiiiectifiable?

4.3. Question. Is it coiisisteiit with Z'FC that every c:oiiiitaLle tleiise iii itself Hniis(lorff spac:e is (:o1111 tní~lv coiiiiectifiable?

4.5. Questioii. Let Al- be a coiiiit¿rble tieiise iii itself Tyt-lioiiofi w-coiiectifinlle c p x e . Does S h v e ; L stroigly c:oiiiiec: tif>iiig f'nniily (see Curolluy 2.3) ?

4 . G . Questioii. Let Al- Le a coiiiitrrble deiise iii itself Tydloiioff' smpeiitiai space. Is theii S coiiiit~rbly coiiiiectifiable?

/

4.9. Questioii. Let S be íi coiiiitalfe tíeiise iii itself Tyc.hoiioE space. Does X Lave a TycJioiioff (theii i:le¿u-ly iiiicoi~iitabk!) (:oiiiiectifit:ntiori?

4.10. Questioii. Let S be n Tyt~ionoffspérce with a coiiiitnbleiietwo1.k a i d without opeii coiiipnct stibspxes. Does S have a Tycliorioff coiiiiectificatioii ?

4.1 1. Questioii. Let ,"L' be iuetric sliacx withotit opeii croiiipact siibspaces. Does X have a Tycdioiioff coiiiiec tificntioii ?

4.12. Questioii. Let S be metric space witlioti t opeii compact stibspaces. Does X have a iuetrizable coiiiiectificntioii?

4.13. Questioii. Let X Le a coiintable dense iii itself Hritis(lorffspace. Does A' have a deiise Syclioiioff stilspace?

4.14. Questioii. Let S be n cormtable (leiise iii itself Haiistlor-ff space. Does X have N I deiise Uiysofiii s i iLspí rcx?

4.16. Question. Let -Y Le íi coiiiitable &.ose iii itself Wnristluiff's1m:e wliicJi has a c 1 eiise Ty di oii o f f s t i Lspr~ce. Is t lieii X co I 11 1 t ably coiiii et: tifiable Y

11

4.17. Questioii. Let S be a coriiitnlle tleiise in itself Haiistloifl space wliicii lins ai w-coiinectifi~iúle tleiise sriLspnce. Is then .Y coiiiitaLly coiiiiec:tifinLle?

4.18. Questioii. Let ,Y Le a coiuitalle deiise iii itself Ifaiistloríf' Uiysoliu space which hczs ai w-c:uiiiiectifiaLle d e u x siil>:;pace. Is theii S <:o1111 taLly coiiiiectifiable?

4.19. Questioii. Let .,Y he n coiiiitable tfeiise iii itself Tythoiioff space which h i i s

í ~ 1 w-cotiuectifiirble tleiise siiúspnce. Is tiieii S coriii tally coiiiiectifiirlle?

4.20. Questioii. Let G' Le a coriutnble iioii-tli.wr-ete Haiistloi-ff topo1ogica.i g-roiip. Is G cotiii taLiy coiiiiec tifinúle?

I

J

J

REFERENCES

[l] Watson S., Wilsoii R.G, Eiulxdtliiigs iii coiiiiected spaces, Hoiistoii 3. Math., 1993, v.19, N , [2] Porter J.R., Woods G.R.., SiilqíLces of coiuectetl spaces, to :~ppea.r

[3J Dow A., Giihli A.V., Szyinaíski A., Itigicl Stoiie sp;~ces witliiii ZFC, Proceed- i ngs of tlie Aiiier. blntli. Soc., 19SS, v.102, N 3, i45-748.

[4) Etla I<., I\:uiio I., Nogiirn T. Spi~ctis, wliicli coiltaiii ; L copy of t l i e rstioiids, Joiirilal of A,J.atli. SOC. of J ~ I > ; u ~ , 1990, 10,3-112.

(51 Rwiiii ME., hiutiii’s íixioiii, iir: Hruirll>ook of hiatlieinntical Logic, .J.Buwise ed., North Hollcuitl, Aiiisterclaiii, 19i7.

[GI Arli,wgt?l’skii A.V., Poiroiiiíuev V.I., &irt?ral to1)oiogy iii prc)l,lziiis a i i t l exercises (iii Riissiui), bíosc:ow, Ní~iiIa. , 1974. [I] Bell M., I<iiiieii I<., 0 1 1 the P I clituactzr of iiltrafiltttrs, C.R. Rlatli. Rep. Acacl. Sci. Cí~liatla, 1951, v.3, N G, 351-356.

ALAS OFELIA T. Iiistitiito cle Mateiiiatica e Estatísticü Uiiiversidade cle Sría Paido Riia do Matáo, 1010 - C.P. 205i0, 01495-970 - SüO Pado, Brasil e-iiiail alas(Qiiiie.itsp.Lr

TKACENKO MII¿IIAIL G. Depar taiiieii t o cle Ma tenis ticas, Uiiiversidatl Aiitóiioiiia Metropoli tailit, Av. Mihoacaii y La Piirísiuia, Iztapalapa, A.P.55-532, C.P.09340, Mexico, D.F. e-iuail illic:ll(oxailiiili. LiíL111 .ills

or rolíLlitlo(~reclvaxl.d~s<:a.iiii¿uii.li~~ to TlaC.eiAo

TKACIIUK VLADIMIR V. Depart amii to tie Ma tenia ticas, UiLiversidacl Au tchoina. Metropoli tiuiíi, Av. blichoacaii y La Piirísiiiia, Iztapalapa, A.P.55-532, C.P.00340, Mesico, D.F. e-iiiail vova~xsiiiiiii.uaii.iiix

13

WILSON RIGIIAILD G. D c p u t ítliieii t o clt! Mat alia t icas , Uuiversitlad Autóiioiiia Metropolitaua, Av. hiíiclioaca y La Piwísinia, Iztapalapa, A.P.55-533, C.P.09340, Mexico, D.F. e-lllail rgw~c3saii i inl . i i~~.l l l~