On the dimension of normal spaces

Abstract

The main purpose of this paper is to generalize for the case of normal spaces the fundamental theorem of the homological dimension theory and the existence theorem on Cantor manifolds in n -dimensional spaces. This generalization is based on (1) The generalizationf of the same theorems, and many others, for bicompacta (i.e. bicompact Hausdorff spaces)—§§ 4, 5. (2) The study of certain ‘maximal’ subspectra of the spectrum of a bicompactum—§ 3. (3) A new approach to the maximal, or Čech, extension of a normal space—§ 6. (4) A suitable definition of the generalized notion of Cantor manifold, which is combinatorial in character and, when applied to compacta, corresponds to the classical definition given by Urysohn—§ 7. In §§ 1,2 I recall well-known definitions regarding homological invariants of bicompacta, and their coverings and spectra (see, for example, my papers (1941 a ,1943) and the fundamental work of Lefschetz (1942, chapter vI)). Notations . U denotes set-theoretical sums, ∩ intersections of sets; ∧ stands for the void set.