A Topological View of Ramsey Families of Finite Subsets of Positive Integers

If is an initially hereditary family of finite subsets of positive integers (i.e., if and G is initial segment of F then ) and M an infinite subset of positive integers then we define an ordinal index . We prove that if is a family of finite subsets of positive integers such that for every the characteristic function χ_F is isolated point of the subspace of { 0,1 }^N with the product topology then for every infinite, where is the set of all initial segments of the members of and ω₁ is the first uncountable ordinal. As a consequence of this result we prove that is Ramsey, i.e., if is a partition of then there exists an infinite subset M of positive integers such that where [M]^{< ω} is the family of all finite subsets of M.