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)H(X) of all auto-homeomorphisms of X  . The pair (X,G)(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(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−1φ(g)=τ∘g∘τ−1 for every g∈Gg∈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)(X,G) be a space-group pair and ∅≠U⊆X∅≠U⊆X be open. We say that U is a small set   with respect to (X,G)(X,G), if for every open nonempty V⊆UV⊆U there is g∈Gg∈G such that g(U)⊆Vg(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}E≠{0}, X is a nonempty open subset of E. For every  X∈NX∈Nthere is a subgroup  GX⊆H(X)GX⊆H(X)such that: (1)  (X,GX)(X,GX)has no small sets, and (2)  {(X,GX)|X∈N}is faithful.