Journal of Mathematical Logic

Under -amalgamation, we obtain the canonical hyperdefinable group from the group configuration.

Let T₁ and T₂ be two countable strongly minimal theories with the DMP whose common theory is the theory of vector spaces over a fixed finite field. We show that T₁ ∪ T₂ has a strongly minimal completion.

Here we prove the existence of sheaf cohomology theory in arbitrary o-minimal structures.

We prove that from categoricity in λ⁺ we can get categoricity in all cardinals ≥ λ⁺ in a χ-tame abstract elementary classe which has arbitrarily large models and satisfies the amalgamation and joint embedding properties, provided and λ ≥ χ.

For the missing case when , we prove that is totally categorical provided that is categorical in and .

This paper gives a formalization of general topology in second-order arithmetic using countably based MF spaces. This formalization is used to study the reverse mathematics of general topology.

For each poset P we let MF(P) denote the set of maximal filters on P endowed with the topology generated by {N_p | p ∈ P}, where N_p = {F ∈ MF(P) | p ∈ F}. We define a countably based MF space to be a space of the form MF(P) for some countable poset P. The class of countably based MF spaces includes all complete separable metric spaces as well as many nonmetrizable spaces.

The following reverse mathematics results are obtained. The proposition that every nonempty G_δ subset of a countably based MF space is homeomorphic to a countably based MF space is equivalent to over ACA₀. The proposition that every uncountable closed subset of a countably based MF space contains a perfect set is equivalent over to the proposition that is countable for all A ⊆ ℕ. The proposition that every regular countably based MF space is homeomorphic to a complete separable metric space is equivalent to over .

We prove that if p ≠ q are distinct primes, then the classification problems for p-local and q-local torsion-free abelian groups of rank two are incomparable with respect to Borel reducibility.