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

Abstract

DefinitionLet 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 1The classK:={(X,H(X))|X is a nonempty open subset of ametrizable locally convex topological vector space E}is faithful.DefinitionLet (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 2Let N be the class of all spaces X such that for some normed spaceE≠{0}, X is a nonempty open subset of E. For everyX∈Nthere is a subgroupGX⊆H(X)such that: (1)(X,GX)has no small sets, and (2){(X,GX)|X∈N}is faithful.