کد مقاله | کد نشریه | سال انتشار | مقاله انگلیسی | نسخه تمام متن |
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4657800 | 1633069 | 2016 | 36 صفحه PDF | دانلود رایگان |
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.
Journal: Topology and its Applications - Volume 210, 1 September 2016, Pages 97–132