A Reconstruction theorem for homeomorphism groups without small sets and non-shrinking functions of a normed space

Let X be a topological space and G be a subgroup of the group H(X) of all auto-homeomorphisms of X. The pair (X,G) is then called a space-group pair. Let K be a class of space-group pairs. K is called a faithfull class if for every (X,G), (Y,H) ∈ K and an isomorphism φ between the groups G and H there is a homeomorphism τ between X and Y such that φ(g) = τ ◦g ◦ τ for every g ∈ G. The first important theorem on faithfulness is due to J. Whittaker [W] (1963). He proved that the class of homeomorphism groups of Euclidean manifolds is faithful. That is, {(X,H(X)) | X is a Euclidean manifold} is faithful. Other faithfulness theorems were proved in R. McCoy [McC] 1972, W. Ling [Lg1] 1980, M. Rubin [Ru1] 1989, M. Rubin [Ru2] 1989, K. Kawamura [Ka] 1995, M. Brin [Br1] 1996, M. Rubin [Ru3] 1996, A. Banyaga [Ba1] 1997, A. Leiderman and R. Rubin [LR] 1999 and J. Borzellino and V. Brunsden 2000.