The product of 〈αi〉-spaces

A proof of Nogura’s conjecture

“…Note that K is an open subset of X [2] , so OCA applies to it. Therefore, to finish the proof, it suffices to show that neither of the following two alternatives given by OCA is possible.…”

“…Case 1. There is uncountable Y ⊆ X such that Y [2] ⊆ K. We may assume that Y actually has size ℵ 1 and since under OCA subsets of N N of size ℵ 1 are bounded in the ordering of eventual dominance (see [8, §8]), going to a subset of Y, we may assume that there is f ∈ N N such that…”

“…This property seems to be considerably less restrictive than the diagonal sequence property as it can be seen, for example, from a result of Nyikos [3] which states that every Fréchet topological group has the weak diagonal sequence property. It turns out that in general the weak diagonal sequence property is not a productive property which motivated T. Nogura [2] to ask if the property is productive under some restriction such as the following. The purpose of this note is to give a positive answer to Nogura's question assuming the Open Coloring Axiom [8, §8].…”

“…Answering a question of T. Nogura (1985), we show using the Open Coloring Axiom that the weak diagonal sequence property is preserved by taking products whenever the products themselves are Fréchet. As an application we show, using the same assumption, that the product of two Fréchet groups is Fréchet provided it is sequential.…”

“…Recall also that a space X is said to be Fréchet if every subset A of X which accumulates to some point x in X contains a sequence which converges to x. It is usually in the realm of Fréchet spaces that one considers various ways to obtain a converging sequence out of a given sequence of converging sequences (see, e.g., [1], [2], [3], [4], [5]). For example, the well-known diagonalsequence property states that if {x nk } is a double-indexed sequence of members of X such that for some x ∈ X and all n, x nk → k x, then for each n we can choose k(n) such that…”