Characterizations of [formula omitted] - or ( Q∞, Σ) -manifolds and their applications

Abstract

We identify Euclidean spaces R n with the subspaces of the countable infinite product R ω . Then the set ⋃ n∈ N R n has two natural topologies, namely the weak topology (the direct limit) with respect to the tower R 1⊂ R 2⊂ R 3⊂⋯ and the relative topology inherited from the product topology of R ω . We denote these spaces by R ∞ and σ , respectively. Thus the bitopological space ( R ∞,σ) is obtained. Replacing R with the Hilbert cube Q=[−1,1] ω , we can define the bitopological space ( Q ∞, Σ) . In this paper, we give several characterizations of the bitopological manifolds modeled on ( R ∞,σ) or ( Q ∞, Σ) , which are applied to bitopological groups, bitopological linear spaces, spaces of measures, spaces of maps, hyperspaces, etc.