Localization in dimension theory

Sullivan (1970, 1974) pointed out the availability and applicability of localization methods in homotopy theory. We shall apply the method to dimension theory and analyze covering dimension and cohomological dimension from the viewpoint. The notion of localized dimension with respect to prime numbers shall be introduced as follows: the P-localized dimension of a space X is at most n (denoted by dimp X ⩽ n) provided that every map f : A → Snp of a closed subset A of X into a P-localized n-dimensional sphere Snp admits a continuous extension over X. in results are: 1. t P1 ⊆ P2 ⊆ P. Then dimp1, X ⩽ dimp2 X (Theorem 1.1). t X be a compactum. Then the following conditions are equivalent: 2.1. mX < ∞; r some partition P1,…, Ps of P, max{dimpi X: i = 1,…, s} < ∞; r any partition P1,…, Ps of P, max{dimpi; X: i = 1,…, s} < ∞ (Theorem 1.2). t X be a compactum, G an Abelian group. We have that sup{c-dimGp X: p ϵ P} = c-dimGX (Theorem 1.4).