Extending Certain Transformation Group Actions in Separable, Infinite-dimensional Frechet Spaces and the Hilbert Cube

A theorem of V. L. Klee, Jr. [9] asserts that any homeomorphism between two compact sets in a separable, infinite-dimensional Hilbert space can be extended to a homeomorphism of the entire space onto itself. Since it has recently been shown ( [ l ] , [5] and [6]) that all separable, infinite-dimensional Frechet spaces are homeomorphic (a Frechet space being a metrizable, complete, locally convex linear topological space), this result holds in all such spaces. A salient property of compact sets in such spaces is that for each nonnull, homotopically trivial open set U, the relative complement in U of a compact set is also nonnull and homotopically trivial (see [3, §§3-5]). This property has been called "property Z" by R. D. Anderson, who has extended Klee's theorem above to the following in [3].