Kernels in dimension theory

Abstract:

All spaces are metrizable. A conjecture of de Groot states that a weak inductive dimension theory beginning with the class of compact spaces will characterize those spaces which can be extended to a compact space by the adjunction of a set of dimension not exceeding n. Nagata has proposed a variant of this conjecture as a means of finding insights into the original conjecture. (See Internat. Sympos. on Extension Theory, Berlin, 1967, pp. 157-161.) The proposed variant replaces compact with $\sigma$-compact. The present paper concerns a study of strong inductive dimension theory beginning with an arbitrary class of spaces. The study is motivated by the above two conjectures. It indicates that a theory of kernels is a more natural by-product of inductive theory than a theory of extensions. An example has resulted which, with the aid of the developed theory and the Baire category theorem, resolves the second conjecture in the negative. The original conjecture is still unresolved. It is also shown that the notion of kernels results in a further generalization of Lelek’s form of the dimension lowering map theorem (Colloq. Math. 12 (1964), 221-227. MR 31 #716).