Compactification and factorization theorems for transfinite covering dimension
Katsuya Yokoi · Tsukuba Journal of Mathematics · 1991
Borst introduced a transfiniteextension of the covering dimension.In thispaper we obtain compactification and factorization theorems for this dimension function. Introduction.In this paper we assume that all spaces are normal.A space X is called weakly infinite-dimensional in the sense of Smirnov, abbreviated 5-w.i.d., if for every sequence {{Ait 5i): z'gN} of pairs of disjoint closed sets in X there is a partition Lt in X between At and Bt for each zeN n such that f ＼ Li=d> for some neN.P. Borst [2] defined a new transfinitedimension function, trdim, by generalizing a necessary and sufficientcondition of n-dimensionality (in the sense of covering dimension) to transfiniteordinals and he classifiedS-w.i.d.spaces by use of the dimension function.This paper is concerned with this dimension function.In section 3 we prove factorization theorem for the above transfinite covering dimension.Recently T. Kimura [6] showed that every space X has a compactification aX of X such that trdim aX<Ltrdim X and w{aX)