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

作者：Fonf Vladimir P^*; Rubin Matatyahu

来源：Topology and Its Applications, 2016, 210: 97-132.

DOI：10.1016/j.topol.2016.04.014

• Homeomorphism group
• Reconstruction
• Locally convex spaces
• Normed spaces
• Uniformly continuous
• Bilipschitz

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) is an element of K and an isomorphism phi between the groups G and H there is a homeomorphism tau between X and Y such that phi(g) = tau (o) g (o) tau(-1) r-1 for every g is an element of G.
Theorem 1. The class K: = {(X, H(X)) | X is a nonempty open subset of a metrizable locally convex topological vector space E} is faithful.
Definition. Let (X, G) be a space-group pair and empty set not equal U subset of X be open. We say that U is a small set with respect to (X, G), if for every open nonempty V subset of U there is g is an element of G such that g(U) subset of 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 not equal {0}, X is a nonempty open subset of E. For every X is an element of N there is a subgroup G(x) subset of H(X) such that: (1) (X, G(x)) has no small sets, and (2) {(X, G(x)) | X is an element of N} is faithful.

• 出版日期2016-9-1