A theorem of dimension theory
Jun-iti Nagata · Proceedings of the Japan Academy Series A Mathematical Sciences · 1956
Recently a dimension theory or general metric spaces has been established by M. Kattov and K. Morita. ) The purpose of this note is to study some necessary and sufficien conditions for n- dimensionality of general metric spaces.In the present note we take Che definition o dimension by H. Lebesgue or that by M. Kattov and K. Morita as the same: dim R=-1 or R=, dim R n if and only if 2or any pair of a closed set F and an open set G with FG here exists an open set U such that FUG, dimB(U) n-l.)Definition.For two collections l[, It' o open sets we denote by 1.(i-1, 2,..., n+ 1) of disjoint collections such that []i=1, ..-, n+ 1; re=l, 2,... is an open basis of R.