β Scribed by Katsuya Yokoi
No coin nor oath required. For personal study only.
Sullivan (1970
Sullivan ( , 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 + S;4 of a closed subset A of X into a P-localized n-dimensional sphere 5'; admits a continuous extension over X.
The main results are:
(1) Let P, & Pz & P. Then dimp, X < dimq X (Theorem 1.1).
(2) Let ,Y be a compactum. Then the following conditions are equivalent: (a) dimX < <co; (b) for some partition PI,.
, P, of P, max{dimp& X: i = 1,. , s} < 00; (c) for any partition PI,.
~ P, of P, max{dimpi X: i = 1,. . , s} < 00 (Theorem 1.2).
(3) Let X be a compactum, G an Abelian group. We have that sup{c-dime, X: p E 'P} = c-dime X (Theorem 1.4). 0 1998 Elsevier Science B.V.
π SIMILAR VOLUMES
Classical dimension theory, when applied to preference modeling, is based upon the assumption that linear ordering is the only elemental notion for rationality. In fact, crisp preferences are in some way decomposed into basic criteria, each one being a linear order. In this paper, we propose that in