Independence, dimension and continuity in non-forking frames

Abstract

The notion $J$ is independent in $(M,M_0,N)$ was established by Shelah, for an AEC (abstract elementary class) which is stable in some cardinal $\lambda$ and has a non-forking relation, satisfying the good $\lambda$-frame axioms and some additional hypotheses. Shelah uses independence to define dimension.

Here, we show the connection between the continuity property and dimension: if a non-forking satisfies natural conditions and the continuity property, then the dimension is well-behaved.

As a corollary, we weaken the stability hypothesis and two additional hypotheses, that appear in Shelah's theorem.