Electronic Version 16 (1991), 45-51

Tatsuo Goto

In 1928, P. Alexandroff proved that a compact subspace X of Euclidean n-space Rⁿ has dimension <= m iff for every compact polyhedron K in Rⁿ of dimension n - m - 1 and every \epsilon > 0 there exists an \epsilon-translation f of X into Rⁿ such that F(X) does not meet K. In this note, we study the metric dimension \mu-dim of bounded subspaces in Euclidean spaces and we shall prove

Theorem. Let X be a bounded subspace in Rⁿ. Then \mu-dim X <= m iff for every compact polyhedron K with dim K = n - m - 1 and every \epsilon > 0 there exists an \epsilon-translation f: X --> Rⁿ such that f(X) and K are disjoint.

• AtlasImage format
• DVI file
• PostScript file

volume 16: table of contents
topology proceedings Electronic Version