Metrizability of Clifford topological semigroups

METRIZABILITY OF CLIFF ORD TOPOLOGICAL SE MIGR OUPS T ARAS BANAKH, OLEG GUTIK, OLES POTIA TYNYK, ALEX RA VSKY Abstract. W e prov e that a topological Cl ifford semigroup S is metrizable if and only if S is an M -space and the set E = { e ∈ S : ee = e } of idempotent s of S is a metrizable G δ -set in S . The same metrization criterion holds also f or any count ably compact Clifford topological semigr oup S . Introduction According to the clas sical Birkhoff-K akutani Theorem [2, 3.3 .12], a topolog ical group G is metriz able if and only if it is first countable. In this paper we establish metrization criteria for topolog ical Clifford semigroups. In pa rticular, in Theo rem 2.4 we pr ov e that a top olog ical Clifford semigro up is metrizable if a nd o nly if S is an M -space a nd the subset of idemp otents E = { e ∈ S : ee = e } is a metrizable G δ -set in S . In Theorem 3.1 we shall prov e that a countably co mpa ct Clifford topolog ical semig r oup S is metrizable if and only if E is a metrizable G δ -set in S . All to po logical spaces co nsidered in this pap er a re regula r . 1. Clifford topolo gical semigroups versus topological Cliff o rd semigr oups In this sectio n we discuss the interpla y b etw een Clifford top ologica l semigro ups and top ologica l Clifford semi- groups. A top olo gic al semigr oup is a topo logical space S endow ed with a con tinuous asso ciative op eration · : S × S → S . A top olog ical semigr o up S is a Cliffor d top olo gic al semigr oup if it is alg ebraically Clifford, i.e., S is the union of groups. F or each element x ∈ S of a Clifford semigr oup there is a unique element x − 1 ∈ S such that xx − 1 x = x , x − 1 xx − 1 = x − 1 and xx − 1 = x − 1 x . This element x − 1 is called the inverse of x . The map ( · ) − 1 : S → S , ( · ) − 1 : x 7→ x − 1 , is called the inversion on S . In Cliffor d to po logical semig roups the inv ersio n is not necessar ily contin uous. By a top olo gic al Cliffor d semigr oup we mean a Clifford top ological semigroup S with con tinuous inversion ( · ) − 1 : S → S . The s implest example o f a Clifford to p o logical semigroup w hich is not a topolo gical C liffo r d semigroup is the half-line [0 , ∞ ) endowed with the op era tion of multiplication o f r eal num b ers. This semigr oup is lo cally co mpact. Suc h examples cannot o ccur a mo ng compac t top olo gical semig roups b eca use o f the following classical r esult whose pro of can b e found in [8] o r [9]. Theorem 1.1. F or e ach c omp act Cliffor d top olo gic al semigr oup S the inversio n op er ation ( · ) − 1 : S → S is c ont inu ous, which implies t hat S is a top olo gic al Cliffo r d semigr oup. F or countably compact Clifford top olo gical semigr o up the inv ers io n op eration is sequentially contin uous. Let us r ecall tha t a function f : X → Y betw een top ological s pa ces is se quential ly c ontinu ous if for each conv ergent sequence ( x n ) ∞ n =1 in X the sequence  f ( x n )  ∞ n =1 is convergen t in Y and lim n →∞ f ( x n ) = lim n →∞ f ( x n ). The following “ countably co mpa ct” v ersio n of Theorem 1.1 was prov ed b y Gutik, Pagon, and Rep ov ˇ s in [7 ]. Theorem 1.2. F or e ach c oun tably c omp act Cliffor d top olo gic al semigr oup S the inversion op er ation ( · ) − 1 : S → S is se quential ly c ontinuous. There ar e also some other conditions guaranteeing that a countably compact Cliffor d top ologica l semigroup is a top olo gical Clifford semigro up. One of such conditions is the top ologica l p er io dicit y . A top ologic a l semigroup S is ca lled top olo gic al ly p erio dic if e ach element x ∈ S is top olo gic al ly p erio dic in the sense that for each neighbor ho o d O x ⊂ S of x there is an integer n ≥ 2 with x n ∈ O x . The pro o f of the following cr iterion can b e fo und in [7]: Theorem 1.3. A c ountably c omp act Cliffor d t op olo gic al semigr oup S is a top olo gic al Cliffor d semigr oup if one of the fol lowing c onditions is satisfie d: (1) the sp ac e S is T ychonoff and the squar e S × S is pseudo c omp act; 1991 Mathematics Subje ct Classific ation. 22A15; 54E35; 54E18; 54D30. 1 2 T ARAS BANAKH, OLEG GUTIK, OLES POTIA TYNYK, ALEX RA VSKY (2) the squar e S × S is c ountably c omp act; (3) S is se quential; (4) S is top olo gic al ly p erio dic and first c oun table at e ach idemp otent e ∈ E . F or in verse Cliffo r d top olog ical semigroups this theor e m was prov ed in [4]. 2. Cardinal chara cteristics of topol ogical Clifford semigroups In this se c tion we ev aluate some cardina l characteristics of top o logical Clifford semigr oups. F or a top olog ical space X and a subset A ⊂ X we shall b e int eres ted in the fo llowing ca rdinal c har a cteristics: • the weight w ( X ) of X , equal to the smallest infinite cardina l κ for which there is a bas e B o f the top olo gy of X with |B | ≤ κ ; • the Lindel¨ of n umb er l ( X ), equa l to the smalle st infinite cardinal κ such that e a ch op en cov er U o f X has a sub cov er V o f cardinality |V | ≤ κ ; • the pseu do char acter ψ ( A, X ) of A in X , e q ual to the smallest ca rdinality |U | of a family U of op en subsets of X suc h that ∩U = A ; • the diagonal numb er ∆( X ) = ψ (∆ X , X × X ), eq ual to the pseudo character of the diago nal ∆ X = { ( x, x ) : x ∈ X } in the square X × X . W e say that a top ological space X has G δ -diagonal if ∆( X ) ≤ ℵ 0 . It is ea sy to s ee that ∆( X ) ≤ w ( X ). By a res ult of Ark ha ngelski [1, I I. § 1], each lo cally co mpact space X has weigh t w ( X ) = l ( X ) · ∆( X ). In particular, each Lindel¨ of lo cally compac t space with G δ -diagona l is metr iz a ble. F or top olo gical inv erse Clifford se mig roups the following theorem was pr ov ed in [3, 2.3(10 )]. Theorem 2.1. F or a top olo gic al Cliffor d semigr oup S and its subset of idemp oten t s E = { e ∈ S : ee = e } we have the upp er b oun d ∆( S ) ≤ ∆( E ) · ψ ( E , S ) . Pr o of. By the definition o f ∆( E ), there is a family U of op en neighbor ho o ds of the diagona l ∆ E in E × E suc h that |U | = ∆( E ) and T U = ∆ E . By a similar r e a son, there is a family V of op en neighborho o ds o f the set E in S such that |V | = ψ ( E , S ) and ∩V = E . It follows from the c ontin uity o f the multiplication a nd the in version on S that for any op en sets U ∈ U and V ∈ V the set W U,V = { ( x, y ) ∈ S × S : ( xx − 1 , y y − 1 ) ∈ U, xy − 1 ∈ V } is open in S × S . Consider the family W = { W U,V : U ∈ U , V ∈ V } a nd obse rve that ∆ S = T W and hence ∆( S ) ≤ |W | ≤ |U | · |V | = ∆( E ) · ψ ( E , S ).  Combining Theorem 2.1 with the equalit y w ( X ) = l ( X ) · ∆( X ) holding for each lo cally compact spa ce X (see [1]), we g et the following Corollary 2.2 . Each lo c al ly c omp act top olo gic al Cliffor d semigr oup S has weight w ( S ) = l ( S ) · w ( E ) · ψ ( E , S ) . This cor ollary implies the following metrizability criter io n: Corollary 2 .3. A Lindel¨ of lo c al ly c omp act top olo gic al Cliffor d semigr oup S is metrizable if and only if the s et of idemp otents E is a metrizable G δ -set in S . In fact, this metr izability criterion holds more gener ally for top olog ical Clifford semigro ups whic h a re M - spaces. T he class of M -spaces includes all metrizable s paces, all Lindel¨ of loca lly compact spac es, and all coun tably compact spac es, see [6, § 3 .5]. Let us reca ll [6, 3.5] tha t a top olo gical s pa ce X is called an M - sp ac e if ther e is a sequence ( U n ) n ∈ ω of o pen co vers o f X such that eac h cov er U n +1 star r efines the cover U n +1 and for an y po int x ∈ X , any sequence x n ∈ St( x, U n ), n ∈ ω , ha s a cluster p oint in X . By a characterization theor em of Morita [10] (see also [6, 3 .6 ]), a top ologica l space X is a n M-spa ce if and only if it admits a closed contin uous map f : X → Y onto a metrizable space Y with countably compact preimages f − 1 ( y ) of p oints y ∈ Y . By [6, 3.8], each M -space with G δ -diagona l is metr izable. This fact co m bined with Theorem 2.1 implies the following metrization theorem for top olo gical Clifford semigro ups. Theorem 2.4. A top olo gic al Cliffo r d semigr oup S is metr izable if and only if S is an M -sp ac e and E is a metrizable G δ -set in S . F or topolo gical inverse Clifford semigr o ups this metriza bilit y criterion w as prov ed in [3]. The following example shows that Theorem 2.4 do es not hold without the M-space assumption. METRIZABILITY OF CLIFFORD TOPOLOGIC AL S EMIGROUPS 3 Example 2.5. Ther e is a non- metrizable c oun t able c ommutative top olo gic al Cliffor d semigr oup S such that its set of idemp otents E is c omp act metrizabl e and op en in S . Pr o of. Let X b e any countable top ologic a l space with a unique non-isola ted p oint x 0 . Define a co ntin uo us semilattice op er ation ∧ on X letting x ∧ y = ( x if x = y x 0 if x 6 = y . Let ˜ X b e the space X endo wed with the top olog y of one-p oint compactification of the countable discrete space X ′ = X \ { x 0 } . Consider the commutativ e Clifford s emigroup S = X × Z endow ed with the topolog y of the topolog ical sum ( X × ( Z \ { 0 } ) ⊕ ( ˜ X × { 0 } ). Her e Z is the dis crete additiv e gr o up o f integers. It is easy to ch eck that S is a top ological Clifford semigr oup whos e set of idemp otents E = ˜ X × { 0 } is metriza ble , compact a nd ope n in S . The semigroup S is metriza ble if and only if so is the space X .  3. Metrizability of count abl y comp act Clifford topol ogical semigr o ups Theorem 2.4 implies that a countably c o mpact to po logical Clifford semigroup S is metriza ble if and only if the set E of idempotents of S is a metr izable G δ -set in S . In this section w e shall g eneralize this metrization criterium to co unt ably co mpact Clifford top ologica l s e migroups. Theorem 3. 1. A c ountably c omp act Cli ffor d top olo gic al semigr oup S is metrizable if and only if the set E of idemp otents of S is a metrizable G δ -set in S . Pr o of. Assume that E is a metrizable G δ -set in S . The contin uit y of the se migroup op eration on S implies that the set of idempo tent s E = { e ∈ S : ee = e } is closed in S and hence is countably compact. Being metrizable, the countably compact space E is compact. Claim 3.2. The sp ac e S is first c ountable at e ach p oint e ∈ E . Pr o of. Using the fact that E is a metriza ble G δ -set in S , it can b e shown that ea ch singleton { e } ⊂ E is a G δ -set in S . Using the regularity of the space S , we can choose a countable decr easing seq ue nce ( U n ) n ∈ ω of op en subsets of S such that T n ∈ ω U n = { e } . The co unt able co mpactness of S implies that for ea ch neighbor ho o d U ⊂ S of e there is n ∈ ω with U n ⊂ U . This means that { U n } n ∈ ω is a neigh b orho o d base at e , so S is firs t count able at e .  Claim 3.3. The inversion ( · ) − 1 : S → S is c ontinuous at e ach idemp otent e ∈ E . Pr o of. By Theorem 1.2, the inv ersio n op er ation is sequentially contin uous at hence is contin uous at each po int x ∈ S having countable neighborho o d base. In particular , the inv ersio n is con tinuous at each idempotent e ∈ E .  Claim 3.4. F or every id emp otent e ∈ E the maximal sub gr oup H e = { x ∈ S : xx − 1 = e } is a metrizable top olo gic al gr ou p. Pr o of. The contin uity of the semigroup o p er ation o n S implies that H e is a par atop olog ic al group. By Cla im 3.3 , the in version of S is con tinuous at the idemp otent e . Co nsequently , the inv ersion o f the para top ological gr oup H e is contin uous a t e and so H e is a to po logical g roup, see [2, 1.3 .1 2]. Since S is fir st countable at e , the paratop olo gical group H e is first coun table. By the Birk hoff-Kakutani Theor em [2, 3.3.1 2], the top olog ical group H e is metrizable.  Claim 3.5. The semigr oup S is t op olo gic al ly p erio dic. Pr o of. T ake any point a ∈ S and denote by A the (clo s ed) set of accumulation p oints of the sequence { a n : n ∈ N } in S . The contin uity o f the semig r oup op era tion implies that A is a clo sed co mm utative subsemig r oup in S . Let π : S → E , π : x 7→ xx − 1 = x − 1 x , denote the pro jection of S on to the set E of idemp otents of S . Obser ve that the map π is not necessar ily con tinuous. W e claim that the pro jection π ( A ) ⊂ E has a minimal element with res pec t to the natura l pa r tial order o n E defined b y x ≤ y iff xy = x = y x . By Zorn Lemma , it suffices to prov e that each linearly ordered subset L of π ( A ) has a low er b ound in π ( A ). F or each element λ ∈ L conside r its low er cone ↓ λ = { e ∈ E : e ≤ λ } = { e ∈ E : λeλ = e } 4 T ARAS BANAKH, OLEG GUTIK, OLES POTIA TYNYK, ALEX RA VS KY and obser ve that it is a closed subset of E . Next, consider the closed subset ↓ L = \ λ ∈ L ↓ λ. The c o mpactness of E implies that each ope n neighborho od of ↓ L contain some lower cone ↓ λ , λ ∈ L . Since ↓ L is a closed G δ -set in E , there is a decreas ing sequence { λ n } n ∈ ω ⊂ L such that ↓ L = T n ∈ ω ↓ λ n . F or every n ∈ ω choose a p oint a n ∈ A with π ( a n ) = λ n and o bserve that λ n a n λ n = ( a n a − 1 n ) a n ( a − 1 n a n ) = a n and hence a n lies in the closed subse t λ n S λ n = { x ∈ S : λ n xλ n = x } of S . Then for every m ≥ n we get λ m ≤ λ n and hence a m ∈ λ m S λ m ⊂ λ n S λ n . By the countable compactness of A , the sequence ( a n ) n ∈ ω has an a ccumu lation p oint a ∞ which lies in the closed subse t T n ∈ ω λ n S λ n of S . Let λ ∞ = π ( x ∞ ) ∈ π ( A ). W e c laim that λ ∞ ∈ T n ∈ ω ↓ λ n = ↓ L , whic h means that λ ∞ is a low er bo und for L in π ( A ). Indeed, for every n ∈ ω , the inclusion a ∞ ∈ λ n S λ n implies λ ∞ λ n = a − 1 ∞ a ∞ λ n = a − 1 ∞ a ∞ = λ ∞ and similar ly λ n λ ∞ = λ ∞ , which means that λ ∞ ≤ λ n . Thu s λ ∞ is a low er b ound of the c hain L a nd b y Zor n L e mma, the set π ( A ) has a minimal element e ∈ E . Let b ∈ A b e any element with π ( b ) = e . W e claim that b is top ologica lly p erio dic. By the co untable co mpactness of S , the seq uence { b n : n ∈ N } has a n accumulation p oint c , which b elongs to the closed subsemig roup A ∩ e S e of S . It follows that eπ ( c ) = e cc − 1 = cc − 1 = π ( c ) and π ( c ) e = c − 1 ce = c − 1 c = π ( c ), whic h mea ns that π ( c ) ≤ e . By the minimalit y o f e in the po set π ( A ), we get π ( c ) = e . Therefore, the p oint c is a n accumulation point of the sequence ( b n ) ∞ n =1 in the metrizable topolo gical group H e . Then be can choose an increasing n umber sequence ( n k ) k ∈ ω such that lim k →∞ ( n k +1 − n k ) = ∞ and the sequence ( b n k ) k ∈ ω conv erges to c . Then the sequence ( b n k +1 − n k ) k ∈ ω tends to the idemp otent e = cc − 1 and lies in the subsemigr o up A . Now we see that the ide mp otent e b elongs to the closed set A of acc umulating po int s of the sequence ( a n ) n ∈ ω . Since S is first countable at e , there is an incr e asing num b er se q uence ( m k ) k ∈ ω such that the sequenc e ( a m k ) k ∈ ω tends to e . By the contin uity of the inv ers io n o per ation at e , the seq uence ( a − m k ) k ∈ ω also tends to e and hence e = ee − 1 = lim k →∞ a m k a − m k = aa − 1 . Since lim k →∞ a m k = e = aa − 1 , the seq uence ( a m k +1 ) k ∈ ω tends to aa − 1 a = a , witnessing that the element a is top ologic a lly perio dic.  Claims 3.2, 3.5 and Theorem 1.3(4) imply Claim 3.6. S is a top olo gic al Cliffor d semigr oup. Being countably compact, the space S is M -space and then it is metriza ble by Theore m 2 .4.  References [1] A.V. Arkhangelski, Structur e and classific ation of top olo gic al sp ac es and c ar dinal invariants , Usp ekhi Mat. Nauk 33 (6) (1978) 29–64 (in Russian). [2] A. Arhangel’skii, M. Tk ac henko, T op olo gic al gr oups and r elate d structur es , Atlan tis Press, P aris; W orld Sci. Publ., NJ, 2008. [3] T. 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Mor ita, Pr o ducts of normal sp ac es with metric sp ac es , Math. Ann. 154 (1964), 365–382. Iv an Franko Na tional University of L viv, Ukraine E-mail addr e ss : t.o.ban akh@gmail.com , ovg utik@yahoo.co m, oles2008 @gmail.com, orav sky@mail.ru