A proof of Nogura’s conjecture

Abstract

Abstract. 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 that the product of two Fréchet groups may not be sequential.Recall that a given topological space X (implicitly assumed to be at least Hausdorff) is sequential if every nonclosed subset A of X contains a sequence which converges to a point outside of A. 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.,. 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 thatThis turns out to be quite a strong property preserved under products and imposing first-countability when X is countable and the topology of X is an analytic subset of 2 X (see [10]). Consider however the weak diagonal-sequence property which asserts that we can make the choice k(n) in such a way that some infinite subsequence of {x nk(n) } converges to x rather than the sequence itself. 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.Question 1 ([2, 3.15]). Suppose X and Y are two Fréchet spaces with the weak diagonal sequence property. Suppose further that their product X × Y is Fréchet. Does X × Y have the weak diagonal sequence property?

Fréchet–Urysohn for finite sets, II

“…There is one case, namely implication 8 in the chart for n ≥ 2, where we know it is both consistent with and independent of ZFC that the implication reverses. The consistent reversal follows from the following result of Todorcevic [19], which answered a question of Nogura [10].…”

Petr Simon (1944-2018)

“…First, [75] he showed that assuming CH such spaces can be constructed, and later in joint work with Tironi, [81], gave partial results in the opposite direction. The final solution to the problem was given by Stevo Todorčević in [120] extending the approach of Simon and Tironi: the Open Colouring Axiom implies a positive answer.…”

The metrization problem for Fréchet groups

“…While Malykhin's problem can only have a consistent positive solution and Theorem 2.1 is a ZFC theorem, the analysis of the combinatorial difficulties seems likely to be similar. The reader is referred to [12] and [16] for applications of OCA which are closely related to the subject matter.…”