Characterizations of (R∞,σ) - or (Q∞,Σ) -manifolds and their applications

We identify Euclidean spaces Rn with the subspaces of the countable infinite product Rω . Then the set ⋃n∈NRn has two natural topologies, namely the weak topology (the direct limit) with respect to the tower R1⊂R2⊂R3⊂⋯ and the relative topology inherited from the product topology of Rω . 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.