On extensions of mappings into $n$-cubes
Shozo Sakai · Proceedings of the Japan Academy Series A Mathematical Sciences · 1968
1. Introduction.The purpose of this note is to give a generali- zation o the results of M. K. Fort, Jr. [1] to the case o arbitrary metric spaces.Let X be a metric space and dim X the covering dimension o X.We denote by I the closed unit cube in Euclidean n-space, where n0.If A is a subset o X and f is a mapping whose domain con- tains A, f is of ype k on A i and only i dim (f-(y) A)=n and f is a mapping o A into I.It will be shown that f can be ex- tended to a mapping (? of X into I such that is o type m-n on X--A.Under the assumption o separability or X, this theorem was proved by A. L. Gropen [2] and essentially by M. K. Fort, Jr. [1].If f is, in addition, of type m-n on A, it will also be shown that the mapping , whose existence is asserted above, is o type m-n on X.These results will be established in 3.The author wishes to express his hearty thanks to Professor K. Morita who has suggested this problem and has given him various valuable advices kindly.2o Auxiliary lemmas We employ the terminology o M. K. Fort, Jr. [1].A finite collection 2: o subsets of a metric space has