Convergence in the Presence of Algebraic Structure

A proof of Nogura’s conjecture

“…The following corollary of the proof of Theorem 2 shows that sequential product or two Fréchet groups is always Fréchet, thus answering Question 6.13(i) of [4].…”

“…This is yet another productive convergence property which strengthens the Fréchet property as well as the weak diagonal sequence property (see, e.g., [4]). …”

“…For example, finding such a product would solve yet another well-known open problem in the area (due to V.I. Malyhin) which asks whether Fréchet property implies metrizability in the realm of countable topological groups (see [4]). While OCA is not sufficient for solving Malyhin's problem it might be relevant to the problem about products of Fréchet groups: Problem 5.…”

“…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…”

The metrization problem for Fréchet groups

“…At the time he posed Problem 1.3, Malykhin was aware of the following consistent counterexample. These questions really ask about the existence of certain filtersknown as FUF filters in the literature -on [ω] <ω (see [7]). Since there is already a recent survey [7] by Shakhmatov on convergence in settings where there is additional algebraic structure, we will focus on a scenario for proving a positive answer to Malykhin's problem and refer the reader to that article for further information.…”