On $κ$-Frechet-Urysohn topological groups

We characterize $κ$-Fréchet–Urysohn topological groups. Using this characterization we show that: (1) a hemicompact topological group is $κ$-Fréchet–Urysohn iff it is locally compact, and (2) if $F$ is a closed metrizable subspace of a topological vector space (tvs) $E$ such that the quotient $E/F$ is a $κ$-Fréchet–Urysohn space, then also $E$ is a $κ$-Fréchet–Urysohn space. Consequently, the product of a $κ$-Fréchet–Urysohn tvs and a metrizable tvs is a $κ$-Fréchet–Urysohn space. Under Martin’s Axiom, we construct a countable Boolean $κ$-Fréchet–Urysohn group which is not a $k_{\mathbb R}$-space.

💡 Research Summary

The paper investigates topological groups and topological vector spaces that satisfy the κ‑Fréchet–Urysohn property, a weakening of the classical Fréchet–Urysohn condition introduced by Arhangel’skii. A space X is κ‑Fréchet–Urysohn if for every open set U and every point x∈U there exists a sequence contained in U converging to x. The authors first give a precise characterization of κ‑Fréchet–Urysohn groups (Theorem 2.1). Three apparently different formulations are shown to be equivalent: (i) a technical condition involving countable families of open sets that converge weakly to the identity, (ii) a more transparent condition that any countable family of neighborhoods of the identity contains a subsequence with a convergent selection, and (iii) the usual definition that the group is κ‑Fréchet–Urysohn. The proof of (iii)⇒(i) is the most delicate; it uses a non‑trivial sequence {pₙ} converging to the identity, balances the weak convergence of the families, and constructs a convergent sequence by “correcting’’ the pₙ‑shifts.

Using this characterization, the authors obtain a complete description of hemicompact groups with the κ‑Fréchet–Urysohn property (Proposition 2.2). For a hemicompact group G the following are equivalent: (a) G is κ‑Fréchet–Urysohn, (b) G is locally compact, and (c) G is a Baire space. The implication (a)⇒(b) is proved by assuming the contrary and constructing a compact set that simultaneously lies inside and outside a prescribed increasing family of compact subsets, leading to a contradiction. The converse uses a known result that locally compact hemicompact groups are κ‑Fréchet–Urysohn. The equivalence with the Baire property follows from standard category arguments.

From Proposition 2.2 the authors deduce Corollary 2.3: a hemicompact group is Fréchet–Urysohn if and only if it is a locally compact Polish group. The proof observes that a Fréchet–Urysohn hemicompact group has countable tightness, which coincides with its character; thus the group is metrizable and, being hemicompact, also separable.

The paper then turns to topological vector spaces. Proposition 2.4 shows that if F is a closed metrizable subspace of a TVS E and the quotient E/F is κ‑Fréchet–Urysohn, then E itself is κ‑Fréchet–Urysohn. The argument selects a decreasing balanced neighbourhood base {Vₙ} in E, uses the quotient map p:E→E/F, and applies Theorem 2.1(ii) to the open sets p(Vₙ∩U). A careful diagonal argument yields a sequence in U converging to 0, establishing the κ‑Fréchet–Urysohn property for E. As a direct consequence (Corollary 2.5) the product of a κ‑Fréchet–Urysohn TVS with any metrizable TVS is again κ‑Fréchet–Urysohn. This contrasts with the general situation for arbitrary spaces, where such products may fail to be κ‑Fréchet–Urysohn (as shown in earlier work).

Motivated by the long‑standing Malykhin problem (whether a countable non‑metrizable Fréchet–Urysohn group exists), the authors pose Problem 2.6: does a countable κ‑Fréchet–Urysohn group exist that is not metrizable? Under Martin’s Axiom they answer this affirmatively in a stronger form. They first recall that a separable zero‑dimensional metrizable space X is a Q‑set iff every function from X to the two‑point discrete group 2 that is a pointwise limit of continuous functions (i.e., belongs to B₁(X,2)) actually coincides with the whole function space 2^X (Proposition 2.7). Using a γ‑set X that is not a Q‑set (such sets exist under MA, see Galvin–Miller), they construct a countable Boolean group G = H ∪ (g+H) where H is a countable dense subgroup of C_p(X,2) and g∈2^X\B₁(X,2). Both H and g+H are Fréchet–Urysohn, and by a theorem of Gabriyelyan they together form a κ‑Fréchet–Urysohn group. However, the characteristic function of H is k‑continuous but not continuous, showing that G fails to be a k_R‑space. This yields Example 2.8: under MA there exists a countable Boolean κ‑Fréchet–Urysohn group that is not a k_R‑space, thereby separating the κ‑Fréchet–Urysohn property from the k_R‑property.

In summary, the paper makes several substantial contributions: (1) it provides an exact, usable characterization of κ‑Fréchet–Urysohn topological groups; (2) it shows that for hemicompact groups the κ‑Fréchet–Urysohn property is equivalent to local compactness and to being a Baire space; (3) it proves that the κ‑Fréchet–Urysohn property is preserved under taking quotients by closed metrizable subspaces and under products with metrizable TVSs; (4) it resolves, under MA, the existence of a countable non‑metrizable κ‑Fréchet–Urysohn group, and demonstrates that such a group need not be a k_R‑space. These results deepen the understanding of sequential-type properties in topological algebra and open new avenues for exploring the interplay between κ‑Fréchet–Urysohnness, metrizability, and other classical topological notions.