Locally convex linear topological spaces that are homeomorphic to the powers of the real line

Abstract

Abstract.We give a characterization of locally convex linear topological spaces that are homeomorphic to the uncountable powers of the real line.

Hyperspace of convex compacta of nonmetrizable compact convex subspaces of locally convex spaces

“…Suppose now that cc(X) is an AR of weight ≥ ω 2 . It easily follows from standard results of Shchepin's theory that there exists a compact convex spaceX of weight ω 2 such that cc(X) is an AR (see [13] and also [4], where the case of locally convex spaces is considered). We may assume that cc(X) = lim ← − cc(S), wherẽ S = {X α ,p αβ ; ω 2 } is an inverse system such that for every α < ω 2 the space cc(X α ) is an AR and for every α, β, β ≤ α < ω 2 , the map cc(p αβ ) is soft.…”

“…We may assume that cc(X) = lim ← − cc(S), wherẽ S = {X α ,p αβ ; ω 2 } is an inverse system such that for every α < ω 2 the space cc(X α ) is an AR and for every α, β, β ≤ α < ω 2 , the map cc(p αβ ) is soft. In its turn, everỹ X α can be represented as lim ← −S α , whereS α = {X αγ ,q α γδ ; ω 1 } is an inverse system in Conv and it follows from the results of Chigogidze [4] that for every α, β, where β ≤ α < ω 2 , the mapp αβ is the limit of a morphism (p γ αβ ) γ<ω1 :S α →S β such that the maps cc(p γ αβ ) are soft and for every γ ≥ δ, γ, δ < ω 1 , the diagram…”

“…The theory of nonmetrizable noncompact absolute extensors which is, in some sense, parallel to that of compact absolute extensors, was elaborated by Chigogidze [4] [5]. One can also consider the hyperspaces of compact subsets in the spaces R τ and conjecture that, for noncountable τ , the hyperspace cc(R τ ) is homeomorphic to R τ if and only if τ = ω 1 .…”