Arhangel'skii, Aleksander V.

Činčura, J.

English

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

Commentationes Mathematicae Universitatis CarolinaeAleksander V. Arhangel'skii; Juraj ČinčuraOn continuous images of almost Tychonoff cubesCommentationes Mathematicae Universitatis Carolinae, Vol. 25 (1984), No. 2, 287--295Persistent URL: http://dml.cz/dmlcz/106301Terms of use:© Charles University in Prague, Faculty of Mathematics and Physics, 1984Institute 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 mustcontain these Terms of use.This paper has been digitized, optimized for electronic delivery andstamped with digital signature within the project DML-CZ: The Czech DigitalMathematics Library http://project.dml.czCOMMENTATIONES MATHEiMATICAE UNIVERSITATIS CAROLINAE 25,2 (1984) ON CONTINUOUS IMAGES OF ALMOST TYCHONOFF CUBES A. V. ARHANGEL'SKII, J. CINCURA Abstract: A class P of Tychonoff spaces is called repreeen-tative (for a class ft) provided that any Tychonoff space (any space of <2 ) is isomorphic with a dosed subspace of a product of spaces belonging to P. For a given class fi of Tychonoff spaces the property of being representative is closely connected with the properties of continuous images of almost Tychonoff cubes with respect to the continuous maps into the spaces of P. In this paper we study the properties of continuous images of almost Ty-chonoff cubes and apply the obtained results to the investigations connected with representativeness of classes of Tychonoff spaces. Key words: Almost Tychonoff cube, dyadic space, almost com-pact spacev almost dyadic space, representative class, weight, tightness, pseudocharacter, caliber, Souslin number, radial space, pseudoradial space, character AMS Subj. Class.; 54B30, 54B99, 54D99 Throughout this paper all topological spaces are supposed to be Tychonoff. IAI denotes the cardinality of A and if A is a sub-set of a space X, then A denotes the closure of A in X. Recall that the tightness t(X) of a space X is the smallest infinite car-' dinal <T such that if x € X, A c X and x e A, then there exists a set B c A with IB! * r and x € B. The pseudocharacter f (X) of a space X is the smallest infinite cardinal T such that for any x E X there exists a collection U of open subsets of X with |UI £ C and fl{U: V € %} « {x}. By w(X) we always denote the weight of X. A subset of a space is said to be canonical closed provided that it is the closure of an open subset of X. /sX de-notes the Cech - Stone compactification of X. If C is a cardinal, then T * denotes the smallest cardinal greater than T-Becall that a space x is said to be almost compact provided that - 287 -the remainder ^X - X contains precisely one point. Obviously, any almost compact space is locally compact. Let T be a cardinal, I the closed unit interval. The subspace Ic - [&\ of the Tychonoff cube 1*̂  where a is an arbitrary point of I is denoted by S<~ and called an almost Tychonoff cube. It is well known that for f > //0 /jS^s r i. e. for <? *• J/Q the space S^ is almost compact. Definition 1. A space X is said to be almost dyadic provided that /)X is a dyadic space and the remainder p X - X contains exactly one point. Clearly, any almost dyadic space is almost compact and for any *£"**#§ S«T is almost dyadic. The space of all countable ordinals is almost compact without being almost dyadic. The following three statements are easy to prove. Proposition 1. If a subspace Y of a space Z is an almost com-pact space, then either Y is closed in Z or Y m fit (and, con-sequently, |/$Y - Yl a 1). Proposition 2. Let X be an almost compact (almost dyadic) space and f: X — * Y a continuous map with f(X) - Y. Then the space Y is either compact (dyadic) or almost compact (almost dyadic). Proposition 3. Let X be an almost compact space, f: X — » Y a continuous map, f(X) • Y and Y be not compact. Then f is a perfect map. Theorem 1. Let X be an almost dyadic space. Then the follow-ing hold: (a) There is an uncoutable closed discrete subspace of X. (b) X is not normal. (c) X contains a canonical closed subspace which is non-me-trizable and dyadic. (d) X contains a subspace homeomorphic with D * (the product of />'.j discrete doubletons). (e) t(X)> h'0 (f) f{X) > /rf0 (not all points of X are of type G^ ). (g) A'i is the caliber of X and, particularly, the Souslin number of the space X is countable (for the definition of caliber see e. g. [1]). (h) X is not collectionwise Hausdorff and not even K^ - weak-ly collectionwise Hausdorff. - 288 -(i) The set of all dim Y where Y is a subspace of X is un-bouded. (j) X is not radial (for the definition of radial spaces see e- g. [1]). (k) X is not countably compact. Proof, (a) Since |/aX - Xl » 1 X is not first-countable at the point p € pX - X. At the same time /*X is a dyadic space and applying [ 7? p. 293, (i)] we obtain that there exists a discrete subspace M of X with |M| s ,<,' such that p is the only accumu-lation point of M in /jX. (b) Take an infinite closed discrete subspace M of X and arbitrary disjoint infinite subsets A. B of M. Then A, B are closed in X and their closures in ft X are not disjoint ( l/sX - X| « « D. (c) The space pX is dyadic and X is an open subspace of /3X< Therefore for each x € X there exists a nei^bourhood V with the compact /|X-closure Fx contained in X. Since Fx is a canonical closed subset of the dyadic space /&X F is a dyadic space. Let for each x e X P be metrizable. Then X is a first-countable dense subspace of ftX so that (see [63 ) /jX is second-countable contradicting (a). Consequently, there exists x e X such that Fx is non-metri .cable. (d) Since any non-metrizable dyadic space contains a subspace homeomorphic with D * (see [7] ) (d) follows from (c). (e) and (f) Since the tightness and the pseudocharacter are mono tonic with respect to arbitrary subspaces and t(D^) > XQf (D*« ) > HQ the statements (e) and (f) follow from (d). (g) Since ftX is a dyadic space the caliber of /sX and any its open subspace is X +. Especially, the Souslin number of X is countable. (h) Immediate from (g) and (a). (i) For any n e N there exists a subspace Y of D * with dim Y t n . (j) It follows from (d) since D ^ is not radial (see [2] ). (k) Immediate from (a). Proposition 4* Let X be an almost dyadic space, f: X — * Y a - 289 -continuous map and the space Y satisfy at least one of the follow-ing conditions: (1) Y is normal (2) Y does not contain an uncountable closed discrete subspace (3) t(Y) £ ,<r0 (4) y(Y) £ /rQ (all points of Y are of type G^ ) (5) Y does not contain a subspace homeomorphic with D ' (6) Y is collectionwise Hausdorff (7) There exists n € N such that for any subspace Z of Y dim Z .$ n. (8) Y is radial (9) Y is countably compact Then f (X) is a dyadic space and the map f is extendable to a continuous map /|X—*Y. Proof. According to Proposition 2 the space f(X) is almost dyadic or dyadic. In the second case the proof is completed. Let f(X) be almost dyadic. Since the properties (1) - (9) are closed--hereditary, by Theorem 1 f(X) ̂  f(X) and then by Proposition 1 f(X) a /s(f(X)) is a compact space. Thus, f can be extended to a continuous map /|X—*Y. In particular, from Proposition 4 it follows: Corollary 1. If a continuous image Y of an almost dyadic space is normal or does not contain an uncountable closed discrete subspace, then Y is a dyadic space. Theorem 2. Let £ be the class of all spaces satisfying at least one of the conditions (1) - (9) of Proposition 4 and X = - n{Xrf : •( € A} a product of spaces belonging to £ . Then no almost dyadic space is homeomorphic with a closed subspace of the space X. Proof* Immediate from Proposition 4 and C 9; 17.2.2.]. Evidently, any topological property which is closed-heredita-ry and possessed by no almost dyadic space can participate in the formulations of Proposition 4 and Theorem 2. Hence, we hare the following general result: Theorem 3. Let P be a class of all spaces which do not con-tain a closed almost dyadic subspace and X m f]{Xd : « e A} a pro-duct of spaces belonging to P. Then no almost dyadic apace (par-- 290 -ticularly, S r for T >^Q) i 8 homeomorphic with a closed subspace of X. By the proof of Proposition 4 it is eTident that if f: X — * Y is a continuous map, X an almost dyadic space and Y has a heredita-ry topological property which is not possessed by any almost dyadic space, then f(X) is a dyadic space. This together with Theorem 1 yield: Theorem 4. Let X be an almost dyadic space, f: X-—*-Y a conti-nuous map with f(X) « Y and one of the following conditions be fulfilled: (1') Y is hereditary normal. (2') Y is hereditary collectionwise Hausdorff. (3') t(Y) £ X 0 . ( O *(T> £ X0> (5') Any discrete subspace of Y is countable. (6') Y does not contain a subspace homeomorphic with D A . Then Y is a compact space with countable base. Proof. By Theorem 1 and the observations preceding Theorem 4 Y is a dyadic space and it is well known that a dyadic space possessing one of the properties (1') - (6') is metrizable. Let 9 and Q. be classes of topological spaces. We shall say that the class P is representative for the class 61 provided that any space belonging to the class Q. is homeomorphic with a closed subspace of a product of spaces belonging to P (in the categorical language - Q. is a subclass of the epireflective hull of P in the category of all Tychonoff spaces). A class P will be said to be representative provided that it is representative for the class of all Tychonoff spaces. Theorem 2 implies now the folloving: Corollary 2. The class £ of all spaces satisfying at least one of the conditions (1) - (9) of Proposition 4 is not represen-tative. Remark. J. Vermeer showed in Cl3l that the class of all nor-mal spaces is not representative. Using the ca .rdinals greater than ,S'0 and #^ we can genera-lize the preceding results in such a way that we obtain rather - 291 -strong necessary conditions for the property of being a represen-tative class. We outline these generalizations. Proposition 5. Let T , A. be cardinals with T =** A * A'0 and f: ^ * Y a continuous map. Then at least one of the following assertions is true: (a) The map f is perfect (and, particularly, f(S,-~) is closed in Y) , f(S^-) contains a closed discrete subspace of the cardinali-ty C (evidently, closed also in Y), f(S«7») contains a topological copy of I and f(S*- ) is almost dyadic (and, consequently, non-normal, non-collectionwise Hausdorff, etc.) (b) f(S^) is a dyadic space and the map f is extendable to a continuous map (&$<> *Y. Proof. If f(S/~ ) is compact, then, evidently, the assertion (b) holds. Let f(S/j.) be not compact. Then applying Propositions 1 - 3 we obtain that f(S*-- ) is almost dyadic and f is perfect. It is well known (see e. g. [7, p. 293, (i)] ) that the space S<~- contains a discrete closed subspace A of the cardinality T*. Then f(A) is a closed discrete subspace of f(Sf-r) and for all y £ f(A) f„^(y) is finite ( f |A is perfect and A is discrete). Consequently, |f(A)|-= I A | « <£* and this implies that the weight w(^Z) of the compact space /s Z where Z *= f(S^ ) is not less than T . At the same time /}Z is a continuous image of I c . E. V. Sfce-pin (E.B. IUenwH [15]) proved that under these conditions w(#Z)^<£* X* implies that /$Z contains a topological copy of the cube I It is easy to see that then Z also contains a topological copy Of I * \ From Proposition 5 it immediately follows: Theorem 5. If a class P of spaces is representative, then for any cardinal T > /fQ there exists a space X~ 6 J* with the following properties: X,-- is not normal, X<^ is not collectionwise Hausdorff, X«- contains a closed discrete subspace of the *2T cardinality T and X/-~ contains a subspace homeomorphic with I . Corollary 3. If a class r is representative and closed—here— y, ti cubes I & ditary, then r contains all discrete spaces and all Tychonoff 292 Remark. If 9 is a left fitting class (i. e. if f: X — * Y is perfect and Y e 99 then X e 9), then using the results of [8] we obtain the following simple characterization of representative-ness: 9 is representative if and only if it contains all almost Tychonoff cubes. The class of all factor spaces of ordered spaces coincides with the class of pseudoradial spaces studied in [2] (for the de-finition see also [1]). The property of being pseudoradial is closed-hereditary and the weight of any pseudoradial dyadic space does not exceed c » 2 ° (see [2]). Hence, I is not pseudo-radial and by Corollary 3 we have: Theorem 6. The class of all pseudoradial spaces is not repre-sentative. Proposition 6. The space u>2 of all ordinals less than w.» (which is countably compact, normal, almost compact) and any non-compact continuous image Y of u>2 contain a compact subspace of the uncountable tightness (and, consequently, a compact sub-space of the uncountable pseudocharacter). Proof. Let f be a continuous map of the space t*-*2 on"to a nott--compact space Y. Then f is perfect and, obviously, |Y| = H?. De-fine a transfinite sequence a: UJ^ + 1 *• Y as follows: a~ is an arbitrary point of Y. For A < to.. + 1 choose a . .. e Y -- ta^ ; f5 £ * J 6UC** t n a t if x € f.-j(aa ..) and ye f.-jU^ ), then x > y. For a limit ordinal cC , 0 <̂  cC £ to , put a^ = f(xw< ) where xu *- sup (U{f.«(a^ ): A <, «* } ). It is easy to check that the subspace X « {x,-.. : f e w. + 1 } of the space Y is isomorphic with the space o>..-f- 1. Theorem 7» Tlie class P of all spaces X such that the tightness of any compact subspace of X is countable is not repre-sentative for the class of all normal countably compact ^paceB. Corollary .4. The class of all spaces with the countable tightness (with the countable pseudocharacter) is not representa-tive for the class of all normal countably compact spaces. Recently, a new cardinal invariant of the type of tightness has been introduced (see e. g. [3]). We write tQ(X) f ,<f0 provided that every real function on X for which the restrictions to all - 2^3 -countable eubepaces of X are continuous is a continuous function on X. V. V. Uspenskii proTed in [12] that if a cardinal T is non-measurable, then t0(I4 )s.F,s'0. Since any space S^ is locally homeomorphic with I l we obtain: Proposition 8. If a cardinal T is non-measurable, then t Q(S r) ^ #y 0. !!!heorem 8. The class & of all spaces X with tQ(X) € />'Q is representative for the class of all spaces of non-measurable car-dinality. Particularly, if there exists no measurable cardinal-then the class <3 is representative. Proof. Let Y be a space of non-measurable cardinality. Then the weight T of Y is also non-measurable and Y can be regarded as a subepace of IT . Obriously, Y m f] 11* - lp}: p 6 I T - Y > and then (see [4; ch. II.,3*53) Y is a closed subspace of the space (1{IT- Ipb P S I T - Yi. But t Q(S r) . h'Q. Let T be an uncountable cardinal. Denote by -̂ - the class of all spaces X with the following property: Any continuous map f: S<£-—*X is extendable to a continuous map /*S^ * X. Then it is easy to see that it holds: Proposition 9« The class ^ is the greatest (with respect to inclusion) productiTe and closed-hereditary class of spaces which does not contain the space S<~. As a corollary of Proposition 9 we obtain: Theorem 10. Let A be a set and for any * € A Ĵ  be a non-representatiTe class of spaces. Then the class P « " U{-£,: * € A} is not representatiTe. The following questions seem to be interesting: (1) For which spaces is the class of all spaces with the countable tightness (pseudocharaoter) representatiTe? (2) Is the class of all spaces with the countable preudocha-racter representatiTe for the class of all spaces with the counta-ble tightness? (3) Is the class of all spaces with the countable tightness representatiTe for the class of all spaces with the countable pseudocharacter? - 294 R e f e r e n c e s [ î ] AFXAHГEJЊCЮIЙ A. B.: Cтpoeнкe к клaccкфккaция тoпoлorнчecкиx пpocтpaнcтв и кapдкнaльныe кнвapкaнтк, Уcпaxм мaтeмaти-чacкиx нay* 33(1978), 29-84 [2] APXAHГEJШCKИЙ A. B.: 0 кeкoтopмx cвoficтвax paдкaлькмx пpocт-paнcтв, Maтeмaткчecкиe aaмeтки, 27(1980), 95-104 [3] AШANGEL'SKII A# V. j Funct ional t i g h t n в s s , Q-spaoes and f -- mb ddings, Coшment. Math. Univ. 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Carol. 2 4 ( 1 9 8 3 ) , 121-154 [13] VEHMEER J . : Two g e n e r a l i z a t i o n s of normal i ty, Wisk. вem. rap . 121, Vrije Uhiv. Amsterdam 1980 [14] VAN ЮER SLOT J . : Some p r o p e r t i e s r e l a t e d t o compactness, MCT 19, Amsterdam 1968 1.5] ЩEШH E. B . : Дoктopcкaя диccepтaция Mocкoвcкий гocyдapcтвeнныfi Katedra a lgebry a t e ó r i e č í -yнивepcитeт им. M.B. Лoмoиocoвa s e l MFF UK мexaннкo-мaтeмaтичacxиfi фaкyдьтeт 842 15 B r a t i s l a v a Mocквa, CCCP Ce koвloven ko (Oblátům 2 3 . 3 . 1984) - 295 

On continuous images of almost Tychonoff cubes

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On continuous images of almost Tychonoff cubes

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