In this paper, we show that a regular space with a locally countable weak-base is g-metrizable. Secondly, we establish the relationships between spaces with a locally countable weak-base (resp. spaces with a locally countable weak-base consisting of א0-subspaces) and metric spaces (resp. locally separable metric spaces) by means of compact-covering maps, 1-sequence-covering maps, compact maps, ...

A topological space has calibre ω1 (resp., calibre (ω1,ω)) if every point-countable (resp., point-finite) collection of nonempty open sets is countable. It has compact-calibre ω1 (resp., compact-calibre (ω1,ω)) if, for every family of uncountably many nonempty open sets, there is some compact set which meets uncountably many (resp., infinitely many) of them. It has CCC (resp., DCCC) if every di...

Consider a measure-preserving action Γ y (X,μ) of a countable group Γ and a measurable cocycle α : X × Γ → Aut(Y ) with countable image, where (X,μ) is a standard Lebesgue space and (Y, ν) is any probability space. We prove that if the Koopman representation associated to the action Γ y X is non-amenable, then there does not exist a countable-to-one Borel homomorphism from the orbit equivalence...

Some results are obtained concerning n(T), the number of countable models up to isomorphism, of a countable complete first order theory T. It is first proved that if n(T) = 3 and T has a tight prime model, then T is unstable. Secondly, it is proved that if n(T) is finite and more than one, and T has few links, then T is unstable. Lastly we show that if T has an algebraic model and has few links...