Aarts, Jan M.

English

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

Toposym 3Jan M. AartsComplementary inductive invariants and dimensionIn: 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. 17--18.Persistent URL: http://dml.cz/dmlcz/700798Terms 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.cz17 COMPLEMENTARY INDUCTIVE INVARIANTS AND DIMENSION J. M. AARTS Delft All spaces under discussion are assumed to be metrizable. Let 0bca non-empty class of spaces which is closed for topological mappings. Then the following topo-logical invariants can be defined. (1) The strong (weak) inductive invariant ^-IndX (^-indjf) induced by the class 0 is inductively defined in a similar way as Ind X (ind X), but starting with the definition that ^-IndX (=0>-indX) = - 1 iff X e0. Of course, inductive dimension (0 = {0}) is the best explored inductive in-variant. The concept of an inductive invariant has been introduced by Lelek [5]. (2) The deficiency of X with respect to 0 is defined as follows: 0-dciX = n if there exists ye 0 such that X c Y and dim Y \ X ^ n. The case that 0 is the class of all compact spaces was first discussed by de Groot [2]. To these invariants we add (3) The surplus of X with respect to 0 is defined by ^-sur X _ n if there exists Ye 0 such that Y a X and dim X \ Y ^ n. 0-defX = n, ^-defZ = oo etc. are defined as usual. E.g. {0}-def X = oo, whenever X # 0. It can be shown quite easily that ^-Ind X ^ ^-sur X for every space X9 if the class 0 is closed monotone (i.e. Ze09 whenever Ye 0 and Z is a closed subset of y). Furthermore, ̂ -Ind X g 0-de(X for every space X9 if the class 0 is closed monotone and open monotone. By Jt(a) and s/(a) we denote the class of all sets of absolute multiplicative and additive Borel class a respectively. (See [4] for definitions. Recall that s/(0) = = {0}, Jf(0) is the class of compact spaces, s/(l) is the class of cr-locally compact spaces [7] and J((l) is the class of topologically complete spaces.) Theorem 1. Let 0 = s/(a) or 0 = Jt(o) for a = 2. Then ^-IndX <J n if and only if there exist Y9Ze0 satisfying Y c= X c Z and dim Z \ Y ^ n.In particular ^-IndX = ^-defX = ^-surJC for every space X. Theorem 2. (See [1].) ^r(l)-Ind X = ^(l)-def X for every space X. Theorem 3. s/(l)-lnd X = s/(l)-s\xr X for every space X. 18 J. M. AARTS Problems. Are the equalities J£(i)-IndX = J((l)-sur X and s/(i)-IndX = = sf(l)-def X valid for every space X ? To prove the second equality for separable spaces is a problem1) posed by Nagata [6]. As follows from the corollary below these equalities are closely related. It is a long unsolved problem whether or not JK(0)--ind X = ^(0)-def X for every separable space X ([2], [3]). Definition. Let 0>, 21 and 0t be topologically closed classes of spaces. ^ and 2, are complementary with respect to St if for every Z e 01 and for all X and Y with X u 7 = Z and X n y = 0 the equality ^-Ind X == .2-Ind Y holds. Theorem 4. -s/(l) awd *^(1) are complementary with respect to Jf(0). s#(<x) and J?(a) are complementary with respect to Jf(l) for a ^ 2. Corollary. / / ^(l)-Ind X == -^(l)-sur X for every separable space X, then s/(i)-IndX = j^(l)-def X for every separable space X. Example. It is known [1] that for the product X of the rationals and the n-dim-ensional cube /", we have ^(l)-Ind X == n. By Theorem 4 it follows that for the product Y of the irrationals and F we have jtf(l)-Indy= n. The proofs of Theorems 1, 3, and 4 will be published in forthcoming papers. References [1] / . M. Aarts: Completeness degree. A generalization of dimension. Fund. Math. 63 (1968), 2 7 - 4 1 . [2] / . de Groot: Topologische studien. Thesis, Groningen, 1942. [3] / . de Groot and T. Nishiura: Inductive compactness as a generalization of semi-compactness. Fund. Math. 58 (1966), 201-218. [4] C. Kuratowski: Topologie I. Warszawa, 1958. [5] A. Lelek: Dimension and mappings of spaces with finite deficiency. Colloq. Math. 12 (1964), 221-227. [6] / . Nagata: Some aspects of extension theory in general topology. Contributions to Extension Theory of Topological Structures (Proc. Sympos., Berlin, 1967). Deutsch. Verlag Wissensch., Berlin, 1969, 157-161. [7] A. H. Stone: Absolute Fa spaces. Proc. Amer. Math. Soc. 13 (1962), 495-499. *) Added in proof: This problem has been solved in the negative by J. M. Aarts and T. Nishiura. 

Complementary inductive invariants and dimension

http://dml.cz/bitstream/handle/10338.dmlcz/700798/Toposym_03-1971-1_5.pdf

Complementary inductive invariants and dimension

Abstract

Similar works

Full text

Available Versions

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