Efimov, B. A.

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Institute of Mathematics AS CR, v. v. i.

Toposym 3Boris A. EfimovOn the imbedding of extremally disconnected spaces into bicompactaIn: Josef Novák (ed.): General Topology and its Relations to Modern Analysis and Algebra,Proceedings of the Third Prague Topological Symposium, 1971. Academia Publishing House of theCzechoslovak Academy of Sciences, Praha, 1972. pp. 103--107.Persistent URL: http://dml.cz/dmlcz/700776Terms of use:© Institute of Mathematics AS CR, 1972Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access todigitized documents strictly for personal use. Each copy of any part of this document must containthese Terms of use.This paper has been digitized, optimized for electronic delivery and stampedwith digital signature within the project DML-CZ: The Czech Digital MathematicsLibrary http://project.dml.cz103 ON THE IMBEDDING OF EXTREMALLY DISCONNECTED SPACES INTO BICOMPACTA B. A. EFIMOV Moskva 1. The formulation of the problem The class of bicompacta decomposes into two classes Stx and 0t2. The first class includes the bicompacta which contain jJN — the Stone-Cech compactification of a countable discrete space — and the second includes the bicompacta which do not contain /?iV. We note that the class dtx contains all the infinite quasi-extremal bicompacta (the closure of any open set of the Fa type is open) and all the dyadic bicompacta of weight ^c = exp H0. (A bicompactum X is called dyadic if X is a continuous image of the Cantor space Dx for some x ^ X0.) The class 0t2 contains all the hereditarily normal spaces (and consequently all the linearly ordered bicom-pacta) and all the sequential bicompacta. 1.1. A necessary and sufficient condition for a bicompactum X to contain fiN is that X can be mapped onto a Tyhonov cube Ic with weight c. Note that pN is an extremally disconnected space (the closure of any open set is open). Moreover, j8JV contains all the extremally disconnected spaces with weight ^c. Naturally there arises a question as to which bicompacta contain extremally disconnected bicompacta having sufficiently large weights? Let x ^ t ^ K0 and denote T1^ = min {m, m* ̂  T}. The cardinal number m will be called admissible if m**° = m. 1.2. Theorem. If a bicompactum X can be mapped onto a Tyhonov cube V, then X contains all the extremally disconnected spaces with weight ^x. Conversely, if a bicompactum X contains an infinite extremally disconnected bicompactum Y, with weight x, and Souslin number t, then X can be mapped onto Jm, m -= xl,i + I where I = exp t if t is accessible or else I = exp cr for any <r < t, if t is a weakly inaccessible cardinal number. The last estimate follows from [1]. 1.3. Theorem. Every infinite extremally disconnected bicompactum X with weight x which satisfies the Souslin condition and for which the cardinal number c + T1/XO is it0-admissible, can be mapped onto V. 104 B. A. EFIMOV 1.4. Theoгem ([2]). Any infinite bicompactum X with weight т and Souslin number t, where т ^ exp exp exp t, contains all extremally disconnected spaces with weight g(expт)+. In [2] a special cardinal number invariant called the strength eX of a topological space X is defined. 1.5. Theoгem ([2]). If a bicompactum X can be mapped onto ľ, then eX ^ т. Conversely, ifeX > т, then X can Ъe mapped onto J log(т+). 2. The imbedding of extremally disconnected spaces as nowhere dense subsets into dyadic bicompacta and their absolutes An absolute pX of a topological space X in the Gleason-Ponomarev sense is an irreducible perfect extremally disconnected preimage of X. According to V. I. Po-nomarev [3] the class of completely regular spaces decomposes into classes of spaces which are co-absolute to one another. X and Y are called co-absolute if pX = pY. top 2.1. Theorem. If a bicompactum X with weight T and satisfying Souslin's condition can be mapped onto JT, then there exists a nowhere dense closed F c pX homeomorphic with pX. A topological space X is called T-dispersed if for every closed F a X there exists a point xeF such that x(x, F) < T, where x(*> F) denotes the character of the point x in F. If, for example, T == K0, then the usual definition of a dispersed space is obtained, i.e. every subset F c X i s not dense in itself. 2.2. Theorem. In order that it may be possible to map a dyadic bicompactum X with weight x onto V it is necessary and sufficient that X be not T-dispersed. 2.3. Theorem [4]. The Continuum Hypothesis is equivalent to the following statement: Every non-metrizable dyadic bicompactum contains /?JV. Denote by (y) the following hypothesis: (y) (VT) (C/(T) = K0) & (Vn) (n < T) => (exp it ^ T) 2.4. Theorem. Let (y) hold. The absolute of any T-dispersed dyadic bicompactum X with weight T can be mapped onto IcxpT. Moreover, in order that a mapping ofpV onto Iexpt may exist it is necessary that TXO = exp T and it is sufficient that cf(r) -= Note that no T-dispersed dyadic bicompactum with weight T can be mapped even onto V. B. A. EFIMOV 105 # 2.5. Theorem. The absolute pX of every non-т-dispersed dyadic bicompactum with weight т contains a closed nowhere dense F cz pX homeomorphic with pX. An analogous statement is true for the absolutes of т-dispersed dyadic bicompacta provided (y) holds. 3. CІasses of non-homogeneous extremally disconnected bicompacta A topological space X is called homogeneous if for any two points x, y є X there exists a homeomorphism ę :X—-> X such that ę(x) = y. As A. V. Arhangelskii has shown [5] every extremally disconnected bicom-pactum with weight c is non-homogeneous. Z. Frolík [б] proved that if theContinuum Hypothesis is true or if there are cardinal numbers in between c and exp c then every infinite extremally disconnected bicompactum is non-homogeneous. The author proved [2] that if cX is weakly accessible and cX ^ log (лwX), then the extre-mally disconnected bicompactum X is non-homogeneous. Here we show the non-homogeneity of some new classes of extremally disconnected bicompacta in a number of cases without using any special set theory hypotheses. These resUlts are a conse-quence of the above theorems and the following Frolík's result [7]: lî E is a closed nowhere dense subspace of an extremally disconnected bicompactum X and if £ contains X (in particular, if E is homeomorphic to X) then E is non-homogeneous. 3.1. Theorem.1) Any extremally disconnected bicompactum X satisfying one of the below conditions is non-homogeneous: 1) X satisfies Souslitťs condition and the cardinal c -f (wX)í/Ho is Ъ$0-admis-sible. 2) X is the absolute of the dyadic bicompactum Y with weight т where Y is not т-dispersed, in particular, if cf(wY) ^ Kj. 3) X is the absolute of a т-dispersed bicompactum with weight т provided (y) holds. 4) X is the absolute of an ordered bicompactum Y wYbeing weakly accessible. 4. The dependence of the power of a bicompactum on its weight Consider |X| as a function which assigns to each space X of weight ^ т the cardinałity of X. If |K| is defined on the set of all infinite metric compacta then it can assume at most two values, viz. K0 or c. This fact is proved independently of the x) Editor's Note. K. Kunen announced in his preliminary report On the eompactification of the integers, Notices Amer. Math. Soc. 17(1970), p. 299, that the usual orderings on the types of ultrafilters are not linear. Now by another theorem in [6] it follows that no infinite extremally disconnected compact space is homogeneous. 106 B. A. EFIMOV Continuum Hypothesis. If \x\ is defined on the set of all bicompacta with weight T then T ^ |x| ^ exp T. A question naturally arises: Is it again possible to show independently of the Generalized Continuum Hypothesis that the function \x\ assumes at most two values? In general, the answer is negative. Namely, in the model M of the set theory ZF in which 1) c = K^, 2) exp c = N^, 3) (Vn) (n < c) => => (exp n < exp c) there exists a bicompactum with weight c for which c < \x\ < < exp c. The existence of such models M is proved by Cohen's method. However, in the case of dyadic bicompacta a positive answer can be given without the GCH or any of its analogues. 4.1. Theorem. The cardinality of any dyadic bicompactum with weight x which is not x-dispersed equals exp T. The cardinality of any x-dispersed dyadic bicompactum equals e = ]T exp nk for some countable sequence nt < n2 < ... < k<G>0 < nk < ... of cardinal numbers, where ]T nk = x. k<coo Note that the cardinal number c is independent of the choice of the sequence 4.2. By nwX we shall denote the -̂weight [3] of a topological space X, i.e. the least cardinal number of a system of open subsets of X cofinal with the system of all open subsets of X ordered according to inclusion. 4.3. Theorem. The cardinality of every bicompactum X co-absolute with a dyadic bicompactum which is not (nwX)-dispersed satisfies the inequalities exp (nwX) = pT| = exp (wX) . and nwX = wX = (nwX)*0. If a bicompactum X co-absolute with a dyadic bicompactum is (7rwZ)-dispersed then the cardinality of X satisfies the inequality £ exp nh ^ \X\ = £ exP exP n* k<o)o k<coo for any sequence of cardinal numbers nx < n2 < ... < nk < ... such that J] n* = = nwX. k<wo Since any two compactifications of a space X are co-absolute the following Corollary gives a negative answer to A. V. Arhangelskii's question as to whether any countable completely regular space S has a compactification of power c. 4.4. Corollary. There exists a countable completely regular space S (for instance a countable dense subset ofIc) such that its every compactification has the power exp c. B. A. EFIMOV 107 References [1] D. A. Vladimirov and B. A. Efimov: On the power of extгemally disconnected spaces and complete Boolean algebras. Dokl. Akad. Nauk SSSR 194 (6) (1970), 1247-1250. [2] B. A. Efimov: Extremally disconnected bicompacta and absolutes. Trudy Moskov. Mat. Obšč. 23 (1970), 235-276. [3] V. I. Pomomarev: On space co-absolute with metric spaces. Uspehi Mat. Nauk 21 (4) (1966), 101-132. [4] B. A. Efimov: Subspaces of dyadic bicompacta. Dokl. Akad. Nauk SSSR 187 (1) (1969), 21-24. [5] A. V. Arhangelskii: An extremally disconnected bicompactum having weight c is non-homo-geneous. Dokl. Akad. Nauk SSSR 175 (4) (1967), 751-754. [6] Z. Frolik: Homogeneity problems for extremally disconnected spaces. Coment. Math. Univ. Carolinae 8 (4) (1967), 757-763. [7] Z. Frolik: Fixed points of maps of extremally disconnected spaces. Proc. Internat. Sympos. on Topology andits Applications (Herceg-Novi, 1968). Savez Društava Mat. Fiz. i Astronom., Belgrade, 1969, 164-167. CEMI AS USSR, MOSCOW 

On the imbedding of extremally disconnected spaces into bicompacta

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On the imbedding of extremally disconnected spaces into bicompacta

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Institute of Mathematics AS CR, v. v. i.