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A New Approach to Dimension

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A New Approach to Dimension Acta Math. Hung. 55 (1--2) (1990), 83--95. A NEW APPROACH TO DIMENSION A. K. O'CONNOR (London) 1. Introduction Early ideas about dimension centred on parametric representations of spaces. The Peano curve however showed that the unit square in the plane has a represen- tation using one parameter. The first rigorous, topological definition of dimension was put forward by Brouwer [1] in 1913. This definition is inductive, assigning di- mension zero to certain disconnected spaces. After giving his definition and showing that it assigns the dimension n to Euclidean n-space R", Brouwer did not develop his ideas further. In 1922/3 the small inductive dimension, denoted by ind was defi- ned for regular spaces independently by Menger [9] and Urysohn [13]. In 1931 (~ech [2] defined the large inductive dimension, denoted by !nd, for normal spaces. The covering dimension, denoted by dim, was defined by Cech [3] in 1933, for arbit- rary topological spaces, based on ideas of Lebesgue concerning the unit cube in R ~. These three definitions of dimension are equivalent for separable metric spaces, see [10, p. 184]. For a detailed account of the development of dimension theory and various approaches to it see [6]. It seems that the pioneers were guided by some intuitive ideas concerning the concept of dimension. They sought a concept that was a topological invariant, that did not assign larger dimensions to smaller sets and such that the n-dimensional unit cube should have dimension n. While they recognized that dimension may increase under continuous mappings they may well have hoped that dimension would not be decreased by continuous, injective mappings. We describe two separable metric spaces with small inductive dimension greater than zero which can be mapped by a continuous, injective map into the Cantor set C, which has small inductive dimension zero. The first example of such a space was given by Sierpifiski [12] in 1921. Let co be the set of all non-negative integers. In 1927 Mazurkiewicz [8] showed that for every nCco there exists a subset X of R" such that ind X=n, but X can be mapped into a space of small inductive dimension zero by a continuous, injective map. These spaces are all separable metric and so the large inductive and covering dimensions behave in the same way. Example 1 which is described briefly below was given by Kuratowski [7] in 1932. It is the graph of a real-valued function defined on C which has the small inductive dimension of the real line rather than that of C despite the fact that the distance in the plane between two points of the graph is greater than the distance between the cor- responding points of C. It would be much more natural if this space was regarded as being zero-dimensional and if in general the dimension of the image of a space under a continuous, injective map was at least as great as the dimension of the space itself. Example 2 described below was given by Erd6s [5] in 1940. It is a subset R0 of 6* 84 A.K. O'CONNOR Hilbert space such that ind R 0 = 1 and there is a continuous, injective mapping of R0 into C. We give a definition of a dimension function, denoted by dm, for T~ spaces for which a continuous, injective map from one space to another does not decrease di- mension. Any non-empty separable space X for which there exists a continuous, injective map f: X-~C satisfies dm X=0. However it is not true that every zero- dimensional, separable space can be mapped continuously and injectively into C as is shown by the space X of Example 3 below. First we require that a space S which can be disconnected between any two dis- tinct points (i.e. given any two distinct points the space can be represented as the union of two disjoint open sets each of which contains one of the points) should have di- mension zero, since any separable space satisfying this condition can be mapped into C by a continuous, injective map. We propose that if a space X can be represented as the union of a countable family of closed sets each with the property of S then it too should have dimension zero. We give an example of such a set X which cannot be disconnected between two of its points (see Example 3). This countable sum property for closed sets is also required for larger dimensions and is central to most of the following results including comparisons with ind and Ind. DEFINITION 1. If A, B are disjoint sets in a topological space X, a subset L of X is said to separate A and B in X if L is closed in X and X",,L= UU W where U, W are disjoint open sets in X such that AcU, BoW. If x, y are distinct points in a topological space and a set L separates {x} and {y} in X we say that L separates x and y in X. DEFINITION 2. Given a T2 space X we obtain Dm X and dm X; each of which is an integer greater than or equal to -1, or o~ as follows: dm X=- 1 if and only if X=0. If nCog, then Dm X<=n if for any pair of distinct points x and y in X there exists a set L which separates x and y in X such that dm L<=n-1; and dm X<-_n if X can be represented as the union of a countable family {Si}F=I of closed subsets of X such that for every i, Dm Si<=n. If n~og, then dmX=n if it is true that dmX<-n whilst it is not true that dm X <- n- 1. dm X=oo if there is no integer n such that dm X<=n. We use the convention Dm X=- 1 if and only if X=0. REMARK 1. The dimension dm of a space is a topological property in the sense that if two spaces are homeomorphic they have the same dimension. REMARK 2. If X is a T~ space, then dm X<= Dm X. PROOF. Clearly we may assume that Dm X is finite and X is non-empty. Let DmX=n, neon. Then writing X= ~J S~ where S~=X and S~=0 for each integer ~=1 i greater than or equal to 2 we see that dm X<=n. Example 3 given below is a space X such that dm X<Dm X. If X is a Tz space then clearly the dimension dm of every subset of Xis defined. It can be shown that Acta Mathematfca Hungarica 55, 1990 .
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