On compactification of metric spaces

Iff:X →X* is a homeomorphism of a metric separable spaceX into a compact metric spaceX* such thatf(X)=X*, then the pair (f,X*) is called a metric compactification ofX. An absoluteG _δ-space (F _σ-space)X is said to be of the first kind, if there exists a metric compactification (f,X*) ofX such that , whereG _i are sets open inX* and dimFr(G _i)]<dimX. (Fr(G _i) being the boundary ofG _i and dimX — the dimension ofX). An absoluteG _δ-space (F _σ-space), which is not of the first kind, is said to be of the second kind. In the present paper spaces which are both absoluteG _δ andF _σ-spaces of the second kind are constructed for any positive finite dimension, a problem related to one of A. Lelek in 11] is solved, and a sufficient condition onX is given under which dim X* −f(X)]≧k, for any metric compactification (f,X*) ofX, wherek≦dimX is a given number. This research has been sponsored by the U.S. Navy through the Office of Naval Research under contract No. 62558-3315.