On the dimension of Hilbert space remainders

Every space is assumed to be separable and metric. A space is called (strongly) countably dimensional if it can be written as a countable union of (closed) nite dimensional subspaces. A space X is called strongly innite dimensional if the space admits an essential system (Fn; Gn) 1 n=1, i.e. Fn and Gn are disjoint closed subsets of X such that if Sn is a closed separator of Fn and Gn for each n, thenT1 n=1 Sn is nonempty. The sequence of left and right endfaces of the Hilbert cube is the standard example of an essential system. A well-known theorem of Engelking [E] states that every autohomeomorphism h of an n-dimensional space X can be extended to a homeomorphism ~h: C! C, where C is an n-dimensional compactication of X (and hence we have a n-dimensional remainder). We consider the question of whether similar results can be obtained for innite dimensional spaces, i.e. is it possible to put a bound on the dimension of the remainder? The following example shows that the answer is no i