Pol, Elżbieta

Pol, Roman

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A hereditarily normal strongly zero-dimensional space containing subspaces of arbitrary large dimension

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

Toposym 4-BElżbieta Pol; Roman PolA hereditarily normal strongly zero-dimensional space containing subspaces ofarbitrary large dimensionIn: Josef Novák (ed.): General topology and its relations to modern analysis and algebra IV,Proceedings of the fourth Prague topological symposium, 1976, Part B: Contributed Papers. Societyof Czechoslovak Mathematicians and Physicist, Praha, 1977. pp. [357]--360.Persistent URL: http://dml.cz/dmlcz/700615Terms of use:© Society of Czechoslovak Mathematicians and Physicist, 1977Institute 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.czA HEREDITARILY NORMAL STRONGLY ZERO - DIMENSIONAL SPACE CONTAINING SUBSPACES OF ARBITRARY LARGE DIMENSION. EL2.BIETA POL AND ROMAN POL WARSZAWA 1. It was an old problem raised by E. Cech [2] whether the covering dimension dim is monotone in the class of hereditarily normal spaces; the analogous problem for the large inductive dimension Ind was investigated by C H. Dowker [3] (cf. also [1] and [12]). Under the assumption of an existence of Souslin's continuum V. V. Filippov [8] solved these problems in the negative exhibiting a hereditarily normal space X with dim X = 0 containing for n = 1,2,,.. a subspace X with dim X = Ind X = n. ' ^ n n n A year ago we constructed [15] using only the usual set theory a hereditarily normal space Z with dim Z = 0 containing a subspace Y with dim Y = Ind Y = 1, and quite recently we improved this construction [16] to get a hereditarily normal space X with dim X = 0 containing subspace X with dim X = Ind X =11 for n = 1,2,... It is worth while to notice that compact hereditarily normal spaces missing the monotonicity of dimensions dim and Ind were constructed recently by V. V. Fedorcuk [5],[6] and A. Ostaszewski [13], under some hypothesis stronger than the continuum hypothesis, and more recently, by V. V. Fedorcuk [7] and E. Pol [14], under the continuum hypothesis. In this note we shall briefly discuss the main idea of our construc-tions. Our topological terminology will follow [4]. 2. B. Knaster and K. Kuratowski gave in their classical work on connectedness [10] the following beautiful construction of a hereditarily disconnected, but not totally disconnected space K : let P be irrationals and Q rationals from the unit real interval I, and let E be a subset of P which is not an F^ - set in P; then define K = Expu ( P N E ) X Q C P X I (compare with the Knaster -Kuratowski Broom [1],[4],[9; p. 22]). The standard proof that dim K = 1 relies on the verification that the sets (P \ E) x {0} and (P\ E) x {1} can not be separated by the empty set. However, one can also observe that each G - set in Px I containing the space K contains a set {p} x I for a point pe P and this property suggests the following construction. Let us split 358 P into n+ 1 disjoint Bernstein sets B .B......B (i.e., each B J o' 1' ' n K m intersects each subspace of P homeomorphic to P [11;§401) , let, for 0 < m < n , R be the set of the points in the n-dimensional cube I exactly m of whose coordinates are rational (i.e., we consider the standard decomposition of ln into n+ 1 zero - dimensional sets [9; p. 19]; note that R° = P and R* = Q) and let us define n m n K - Un B x R . Then each G - set in P x I containing K n m=0 m n a n contains also a set {p> x I for a point peP and hence dim K = n. We have seen that the dimension of the spaces K or K was designated by the Borel properties of the sets E or B , respectively. Following this observation, in ojrder to obtain a perfectly normal space with local dimension less than dimensions dim or Ind, (which in fact is just what we need) we would like to get through constructions analogous to that of K or K with an appropriate perfectly normal space B, in which local Borel properties of sets differ from global, instead of P. We shall describe a correspondent space B in the next section. 3. Let -(a. be the set of all countable ordinals with the discrete N topology, let N be the set of natural numbers and let B()£-) = io be the Baire space of weight )f . To define the space B we give the N set oj a topology finer than the topology of B(xv), by taking as a N base the sets Ufl {x6_ : x(n) < a for n G N}, where U is an open set in the Baire space B()f.) and a < co . The space B is perfectly normal, but not paracompact. Although B is locally second - countable and thus its local structure is quite different from those of B()f ), the topology of B is closely related to the topology of B()f_) and, in particular, the Borel sets of B and of B()f_ ) coincide. The detail informations about B the reader can find in [17]. 4. A. H. Stone [19] defined a non - Borel subset E of the Baire space B()f, ) such that each separable subspace of E is countable. The set E considered as the subspace of B is also non-Borel, however E is locally countable in B . We let (compare with the definition of K in sec. 2) Y = Expu ( B V E ) x Q c B x i . Since E is locally countable, we infer from the sum theorem that the local dimension loc dim Y = 0 , whereas dim Y = Ind Y = 1 , by the same arguments as in case of K . The space Z we are looking for can be now defined as follows : let 359 Z = Y u {p} where p£Y , Y is an open subspace of Z and the basic neighbourhoods of the point p are taken in the form {p} U U , where the space Y \ U is open - and - closed in Y , second countable and zero - dimensional. One can easily verify that the space Z is hereditarily normal, Lindelof and dim Z = 0 . 5. The construction of the spaces X will imitate the definition of the spaces K given in the section 2, where the space B (sec. 3) will be used in place of irrationals P . For the purpose we shall exploit the decomposition B ,...,B of the Baire space B(vj* ) into n+1 disjoint non - separable analogues to the Bernstein sets in P, constructed in [18]. The sets B considered in the space B are locally Fa - sets, but their global properties are similar to those of the Bernstein sets, n We let X = U (B xRra)c B x in . n .^-r. m n m=u By the sum theorem we have loc dim X = 0 . However, as in the n * case of the space K , each G - set in B x I containing the space X , contains a set {p} x i for a point pGB and this easily implies that the projection of X onto the n - dimensional cube In is an essential mapping (cf. [1]). Thus n<dim X <Ind X , while Ind X < Ind B + Ind In = n . n — The hereditarily normal space X with dim X = 0 containing all of the spaces X can be obtained in the same way as the space Z from the preceding section, where we let Y to be the free union of the spaces X . REFERENCES. [1] P.S.Aleksandrov, B.A.Pasynkov, Introduction to ̂ dimension theory (in Russian), Moskva 1973. [2] E.£ech, Problem 53, Coll. Math. 1 (1948), 332. [3] CH.Dowker, Local dimension of normal spaces, Quart. Journ. Math. Oxford 6 (1955), 101-120. [4] R.Engelking, General topology, Warszawa 1976. [5] V.V.Fedorcuk, Compatibility of some theorems of general topology with the axioms of set theory (in Russian), DAN SSSR 220 (1975), 786-788. [6] V.V.Fedorcuk, Fully closed mappings and compatibility of some theorems of general topology with the axioms of set theory (in Russian), Mat. Sb. 99 (1976), 3-33. 360 [7] V.V.Fedorcuk, On the dimension of hereditarily normal spaces (preprint). [8] V.V.Filippov, On the dimension of normal spaces (in Russian), DAN SSSR 209 (1973), 805-807. [9] W.Hurewicz, H.Wallman, Dimension theory, Princeton 1941. [10] B.Knaster, K.Kuratowski, Sur les ensembles connexes, Fund. Math. 2 (1921), 206-255. [11] K.Kuratowski, Topology, vol. I, New York 1966. [12] K.Nagami, Dimension theory. New York 1970. [13] A.Ostaszewski, A perfectly normal countably compact scattered space which is not strongly zero - dimensional (preprint). [14] E.Pol, Remark about Juhasz - Kunen - Rudin construction of a hereditarily separable non - Lindelof space. Bull. Acad. Pol. Sci. (to appear). [15] E.Pol, R.Pol, A hereditarily normal strongly zero - dimensional space with a subspace of positive dimension and an N - compact space of positive dimension, Fund. Math, (to appear). [16] E.Pol, R.Pol, A hereditarily normal strongly zero - dimensional space containing subspaces of arbitrary large dimensions, Fund. Math, (to appear). [17] R.Pol, A perfectly normal, locally metrizable, non - paracompact space, Fund. Math, (to appear). [18] R.Pol, Note on decompositions of metrizable spaces II, Fund. Math, (to appear). [19] A.H.Stone, dn a-discretness and Borel isomorphism, Amer. Jour. Math. 85 (1963), 655-666. DEPARTMENT OF MATHEMATICS AND MECHANICS, UNIVERSITY, 00-901 WARSAW (WYDZIAL MATEMATYKI I MECHANIKI, UNIWERSYTET WARSZAWSKl) 

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A hereditarily normal strongly zero-dimensional space containing subspaces of arbitrary large dimension

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