About remainders in compactifications of paratopological groups

ABOUT REMAINDERS IN COMP A CTIFICA TIONS OF P ARA TOPOLOGICAL GR OUPS FUCAI LIN AND SHOU LIN Abstract. In this paper, we prov e a dichotom y theorem for remainders in compactificat ions of paratopological groups: ev ery remainder of a paratopo- logical group G is either Lindel¨ of and meager or Bair e. Moreov er, we give a negativ e answ er for a question p osed by D. Basile and A. Bella in [6] , and some questions ab out remainders of paratopological groups are p osed in the pap er. 1. Introduction By a remainder of a space X we understand the subs pace bX \ X of a Hausdorff compactification bX of X . Remainders in compactifications of top olog ical spaces hav e been studied by some topologis ts in the last few years. A famous clas sical result in this study is the following theorem of M. Henriksen and J. Isb ell [9]: (M. Henri ksen and J. Isb ell ) A space X is of countable type if and only if the remiander in any (in some) compactification of X is Lindel¨ of. Since topo logical gr o ups are muc h more sensitive to the pro p er ties of their re- mainders than top ological spaces in genera l, top ologists are mainly interesting in the remainder s of top olog ical groups or para top o logical groups. F or instance, Arhangel’ski ˇ ı has re c e n tly proved the following tw o dichotomy theorems ab out re - mainders in co mpactifications of top o logical groups: Theorem 1.1. [1] If G is a top olo gic al gr oup, and s ome r emainder of G is n ot pseudo c omp act, then every r emainder of G is Lindel¨ of. Theorem 1.2. [2] Supp ose that G is a non-lo c ally c omp act t op o lo gic al gr oup. Then either every r emainder of G has the Bair e pr op ert y, or every r emainder of G is σ - c omp act. Moreov er, D. Basile a nd A. Bella hav e just shown a dichotom y theo rem for homogeneous spa ces: Theorem 1.3. [6 ] The re mainder of a homo gene ous sp ac e is either Bair e or m e ager and r e alc omp act. D. Basile and A. Bella p osed the following questio n. Question 1.1 . [6] Let X b e a ho mogeneous space and le t bX b e a compa ctification of X . Is it true that the remainder bX \ X is e ither pse udo c o mpact or r ealcompact and mea ger? 2000 Mathematics Subje ct Classific ation. 54A25; 54B05; 54D35; 54D40. Key wor ds and phr ases. Remainders in compactifications; paratop ological groups; top ol ogical groups; homogeneous spaces; Bair e s paces; meager spaces; Lindel¨ of; k -gen tle. Supported by the NSFC (No. 10971185) and the Educational Departmen t of F uj i an Province (No. JA09166) of China. 1 2 FUCAI LIN AND SHOU LIN In [6], D. Basile a nd A. B ella has s hown that none o f the Arhangel’ski ˇ ı’s di- chotom y theorems can b e genera lized to the c a se of homogeneous spaces. In [4], Arhangel’ski ˇ ı ha s given an example to show that the Theore m 1.1 can not b e gener- alized to the case of paratop olo gical gro ups. Natur ally , the following tw o q uestions arise: Question 1.2 . How ab out the dichotomy theor em of the r emainders in co mpactifi- cations of pa ratop olog ical groups? Question 1.3 . Can the dichotom y theo r em 1.2 be generalized to the cas e of pa ratop o- logical g roups? In this pa per , we show that, for a par atop ologica l gr oup G , every remainder of G is either Lindel¨ of and meager or Baire, which g ive an answer for Question 1.2. Also, we give a partial answer for the Q uestion 1.3. Finally , w e give a neg ative answer for the Question 1 .1. 2. Preliminaries Recall that a top olo gic al gr oup G is a g roup G with a top ology such that the pro duct map of G × G into G asso cia ting xy with ar bitrary ( x, y ) ∈ G × G is jointly contin uous a nd the inv erse map of G onto itself ass o ciating x − 1 with arbitra ry x ∈ G is co n tinuous. A p ar atop olo gic al gr oup G is a g r oup G with a top ology such that the pro duct map of G × G in to G is jointly contin uous. A semitop olo gic al gr oup G is a group G with a to p olo gy such that the pro duct map of G × G into G is separa tely contin uous . A quasitop olo gic al gr oup G is a gro up G with a top olog y such that it is a semitop olo gical g r oup a nd the inv erse map of G onto itself is contin uous. Recalled that a space is Bair e if the in tersectio n of a sequence of op en and dens e subsets is dense. Moreover, a space is called me ager if it ca n b e represented as the union o f a seq uence of nowhere dense subsets. Let us call a map f of a space X in to a space Y k -gentle [4] if for every compact subset F of X the image f ( F ) is a lso compac t. A semitop ologica l group G will b e called k -gentle [4] if the inv erse map ( x 7→ x − 1 , ∀ x ∈ G ) is k - g ent le. A family A of op en subsets of a space X is ca lled a b ase of X at a s et A if A = ∩A and for any neighbor ho o d U o f A , there is a V ∈ A such that A ⊂ V ⊂ U . If A is countable, then we s ay that A has countable character in X . A space X is of c ountable typ e [8 ] if every compa ct s ubspace F of X is contained in a compact subspace K ⊂ X with a co un table base of op en neighborho o ds in X . Throughout this pap er, a ll spaces are assumed to b e Tyc honoff. Denote pos i- tively natural n umber by N . W e refer the reader to [3 , 8] fo r notations and termi- nology no t explicitly g iven here. 3. Remainders of p ara topological gr oups Firstly , we give a lemma. Lemma 3 .1. [4] L et G b e a p ar atop olo gic al gr oup. If ther e exists a non-empty c omp act subset of G of c oun table char acter in G , then G is of c ountable typ e. Now, we give a dichotom y theorem o f the rema inder s in compactifications of paratop olo gical g roups. ABOUT REMAINDERS IN COMP ACTIFICA TIONS OF P ARA TOPOLOGICAL GROUPS 3 Theorem 3.1 . L et G b e a non-lo c al ly c omp act p ar atop olo gic al gr oup. Then either every r emainder of G has the Bair e pr op erty, or every r emainder of G is me ager and Lindel¨ of. Pr o of. Supp o se that bG is a compactification of G such that the remainder Y = bG \ G do es not hav e the Ba ire pr o pe rty . Next, we shall prov e that Y is Lindel¨ of and mea ger. Since Y do es not have the Baire pr op erty , there exists a coun table family { U n : n ∈ N } of ope n subse ts of Y suc h tha t ∩{ U n : n ∈ N } is not dense in Y . Because G is no where lo cally compact, Y is dense in bG . F or each n ∈ N , there exists an op en subset V n of bG such that U n = V n ∩ Y . Le t γ = { V n : n ∈ N } . Ther efore, we can find a non- empt y op en subset U of bG such that ( ∩ γ ) ∩ ( U ∩ Y ) = ∅ . It follows that the subspace Z = ( ∩ γ ) ∩ ( U ∩ G ) = ( ∩ γ ) ∩ U is ˇ Cech-complete in U ∩ G of G by [8, Theor em 3.9.6]. It is k nown that every ˇ Cech-complete space is of countable t yp e. Since Z is ˇ Cech-complete, there e xists a non-empty co mpact subset F o f Z o f countable character in Z . Because Z is dense in the op en subspac e U ∩ G of G , F is of countable ch ara cter in U ∩ G [3]. Beca us e U ∩ G is o pe n in G , F is of co un table character in G . Obviously , F is compact in G . Therefore, G is of countable type by Lemma 3.1. Therefor e, Y is Lindel¨ of by M. Henrik sen and J. Isbell theorem. Moreov er, Y is meager by Theor em 1.3. This complete the pro of.  Remark Observe that a r emainder Y o f a non-lo ca lly compact paratop olog ical group G cannot hav e the Bair e prop erty , b e Lindel¨ of and meager at the same time. Indeed, it is easy to see that the failure of the Ba ire prop erty is equiv a lent to the existence of so me non- e mpt y op en meag er subset. Thus we hav e the following t wo corolla r ies. Corollary 3.1. Le t X b e a n either Bair e nor me ager sp ac e. Then X c annot b e a r emainder in c omp actific ations of any p ar atop olo gic al gr oup. Corollary 3.2. L et X b e a neither Bair e n or Lindel¨ of sp ac e. Then X c ann ot b e a r emainder in c omp actific ations of any p ar atop olo gic al gr oup. Remark D. Basile and A. Bella has shown that there exists a homo g eneous space such that the remainder of so me compa ctification is neither B aire nor Lindel¨ of, see [6, E xample 3.3]. H ence Theo rem 3.1 ca n not b e g eneralized to the ca s e of homogeneous spa ces. How ever, w e hav e the following questio n. Question 3.1 . Let X b e a non-lo ca lly compact semitop ologica l group or quasitop o- logical gro up, a nd let bX b e a co mpactification of X . Is it true that the remainder bX \ X has the Baire prop erty or is Lindel¨ of a nd meag er? Next, we obtain tw o corollar ies from Theorem 3 .1. Firstly , we show that the Arhangel’ski ˇ ı’s dichotom y Theor e ms 1.2 can b e g e ne r alized to the case of k -gentle paratop olo gical g roups, which give a pa rtial answer for Q uestion 1.3. Lemma 3.2. [4] L et G b e a k -gentle p ar atop olo gic al gr oup such t hat some r emainder of G is Lindel¨ of. Then G is a top olo gic al gr oup. Corollary 3.3. L et G b e a non-lo c al ly c omp act k -gent le p ar atop olo gic al gr oup. Then either every r emainder of G has the Bair e pr op erty, or every r emainder of G is σ -c omp act. 4 FUCAI LIN AND SHOU LIN Pr o of. Supp o se that bG is a compactificatio n of G , and put Y = b G \ G . By Theorem 3.1, Y has the Baire prop erty , or is meager and Lindel¨ o f. Suppo se that Y do es not hav e the Bair e prop er ty . Then Y is Lindel¨ of, and hence G is a top olog ical group by Lemma 3.2. Then Y is σ -compac t b y Theorem 1.2.  It follows from [1] that a remainder in some compactification of a top ologic al group is metacompact iff it is Lindel¨ o f iff it is r ealcompact. Therefor e, we hav e the following q uestion. Question 3.2 . Assume that G is a non-lo cally compact par atop ologica l gro up, and put Y = bG \ G . Are the following conditions equiv alent? (1) Y is metacompa ct; (2) Y is Lindel¨ o f; (3) Y is rea lcompact. A space X is called metac omp act if ea ch op en covering of X can b e refined by a po in t-finite open cov ering. A space X is ca lled c c c if every disjoint family of op en subsets o f X is countable. Lemma 3.3. [7] Every p oint-finite op en c ol le ction in a c c c Bair e sp ac e is c ountable. The next coro llary gives a partial answer for the Question 3.2. Corollary 3.4. Assume t hat G is a non-lo c al ly c omp act p ar atop olo gic al gr oup, and put Y = bG \ G . If Y is metac omp act and c c c, then Y is Lindel¨ of. Pr o of. By Theorem 3.1, Y ha s the Bair e prop erty , or is meager and Lindel¨ o f. Sup- po se that Y has the Baire prop erty . Then Y is Lindel¨ o f by Lemma 3.3. Hence Y is Lindel¨ of.  Now, w e shall give a nega tive answer for Questio n 1.1 by Exa mple 3.1. Example 3.1. There exis ts a pa ratop olog ic al gro up X such that some compacti- fication bX of X has a rema inder which is neither pseudo co mpact nor mea ger. Pr o of. Let Z = X ∪ Y be the tw o- a rrows space of P . S. Alexandroff and P . S. Urysohn [8, Exercise 3 .1 0. C], wher e X = { ( x, 0) : 0 < x ≤ 1 } and Y = { ( x, 1) : 0 ≤ x < 1 } . The space X is the a rrow spa c e which is homeomorphic to the So rgenfrey line, see [8, Exa mple 1. 2 . 2]. Z is a Hausdorff co mpactification of Sorg e nfr ey line X , and its remainder Y is still a copy of So rgenfrey line. Moreov er, ther e exists a natural structure o f an Ab elian gro up on Y s uch that the multiplication ( u , v ) 7→ u · v is contin uous, that is, the space Y admits a structure of a paratop ologica l group. F or example, if u = ( x, 1) a nd v = ( y , 1) are tw o po int s in Y , then u · v = ( x + y, 1) if x + y < 1, a nd u · v = ( x + y − 1 , 1) if x + y ≥ 1. How ever, Sorg enfrey line is no n-pseudo compact; otherwise, Sorge nfrey line is a compact space since it is a Lindel¨ of space, which is a contradiction. Moreover, since X has the Ba ire prop erty [5], X is non-meager . Ther efore, Y is neither pseudo compact nor meager .  Remark It follows from Example 3.1 that, in Question 1.1, the answer is also negative if we repla ce the “ homogeneous spa ce” by “pa ratop olog ical gr oup”. ABOUT REMAINDERS IN COMP ACTIFICA TIONS OF P ARA TOPOLOGICAL GROUPS 5 References [1] A. Arhangel’ski ˇ ı, Two typ es of r emainders of top olo gic al gr oups , Comment. Math. Univ. 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Isb ell, Some pr op erties of c omp actific ations , Duk e Math. J., 25 (1958)8 3– 106. Fucai Lin(corresponding author): D ep ar tmen t of Ma thema tics a nd Informa tion Sci- ence, Zhan gzhou Normal University, Zhang zhou 3630 00, P. R. China Fucai Lin: Dep a r tment of Mat hemat ics, Sichuan University, Chengdou, 61 0064, P.R.China E-mail addr ess : linfucai2 008@yahoo.co m.cn Shou Lin: Instit ute of Ma thema tics, Ningde Teachers’ College, Fujian 352100 , P. R. China E-mail addr ess : linshou@p ublic.ndptt. fj.cn