A reconstruction theorem for locally convex metrizable spaces, homeomorphism groups without small sets, semigroups of non-shrinking functions of a normed space

Definition 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⁻¹ for every g∈G. Theorem 1 The class K:={(X,H(X))|X is a nonempty open subset of ametrizable locally convex topological vector space E} is faithful. Definition Let (X,G) be a space-group pair and ∅≠U⊆X be open. We say that U is a small set with respect to (X,G), if for every open nonempty V⊆U there is g∈G such that g(U)⊆V. Remarks (a) We do not know whether the members of K have small sets. (b) Earlier faithfulness theorems, required the existence of small sets. Theorem 2 Let N be the class of all spaces X such that for some normed space E≠{0}, X is a nonempty open subset of E. For every X∈N there is a subgroup G_X⊆H(X) such that: (1) (X,G_X) has no small sets, and (2) {(X,G_X)|X∈N} is faithful.