A SURVEY OF DIMENSION THEORY

Abstract

This chapter presents a survey of modern dimension theory emphasizing the development since 1961 when the first Prague Symposium was held. The chapter discusses the dimension theory of metric spaces and non-metrizable spaces. As M. Katetov and K. Morita extended principal results of the classical dimension theory like sum theorem, decomposition theorem and product theorem to general metric spaces and proved dim R = Ind R for every metric space R, there has been a remarkable progress in the theory for metric spaces. Comparing the general imbedding theorem with the classical one for separable metric spaces, it is noticed that P(A) has infinite dimension while every n-dimensional separable metric space is imbedded in the (2n + l)-dimensional Euclidean cube I2n + l. In contrast to the metric case, there are many fundamental problems to be solved in the nonmetrizable case. For example, it is not known whether ind R = Ind R for every compact space R though it is not true for normal spaces. The chapter presents a list of grade to show how good or bad the present status of dimension theory for nonmetrizable spaces is in comparison with the theory for separable and nonseparable metric spaces.