Baumgartner introduced the class of partial orderings for Axiom A which includes c.c.c. p.o. sets, ω 1 -closed p.o. sets and various notions of forcing which add new subsets of ω. If partial orderings which satisfy Axiom A are iterated under countable support, then the iteration, regardless of its length, satisfies a certain covering property. This covering property implies that ω 1 is preserved. It is not plausible, however, that the iteration itself satisfies Axiom A. In this paper we generalize the class of partial orderings for Axiom A so that our generalization is iterable under countable support