Baire-type properties of topological vector spaces

Burzyk, Kliś and Lipecki proved that every topological vector space (tvs) $E$ with the property $(K)$ is a Baire space. Kcakol and Sánchez Ruiz proved that every sequentially complete Fréchet–Urysohn locally convex space (lcs) is Baire. Being motivated by the property $(K)$ and the notion of a Mackey null sequence we introduce a property $(MK)$ which is strictly weaker than the property $(K)$, and show that any locally complete lcs has the property $(MK)$. We prove that any $κ$-Fréchet–Urysohn tvs with the property $(MK)$ is a Baire space; consequently, each locally complete $κ$-Fréchet–Urysohn lcs is a Baire space. This generalizes both the aforementioned results. We construct a feral Baire space $E$ with the property $(K)$ and which is not $κ$-Fréchet–Urysohn. Although a $κ$-Fréchet–Urysohn lcs $E$ can be not a Baire space, we show that $E$ is always $b$-Baire-like in the sense of Ruess. Applications to spaces of Baire functions and $C_k$-spaces are given.

💡 Research Summary

The paper investigates Baire-type properties of topological vector spaces (tvs) by introducing a new, weaker condition called property (MK), which sits strictly below the classical property (K). A null sequence (xₙ) in a tvs is a K‑sequence if every subsequence contains a further subsequence whose series converges; property (K) requires that every null sequence be a K‑sequence. The authors replace “null” by “Mackey null” – a sequence for which there exists an increasing unbounded scalar sequence (aₙ) with aₙxₙ → 0 – and define property (MK) as the requirement that every Mackey null sequence be a K‑sequence.

First, they establish that (K) ⇒ (MK) (Proposition 2.1) and that any locally complete locally convex space (lcs) possesses (MK) (Proposition 2.2). The proof uses the locally complete structure to turn the Mackey null sequence into a bounded set inside a compact absolutely convex hull, then works inside the Banach space generated by that hull to obtain convergence of the associated series. This shows that local completeness is sufficient for (MK), even though it does not guarantee (K) (as Example 2.4 (ℓ₂) demonstrates).

Next, the authors consider a broader class of spaces: κ‑Frechet–Urysohn spaces, introduced by Arhangel’skii. A space X is κ‑Frechet–Urysohn if for every open set U and every point x∈U there exists a sequence from U converging to x. This is strictly weaker than the usual Frechet–Urysohn property. The central result (Theorem 1.10) proves that any κ‑Frechet–Urysohn tvs with property (MK) is a Baire space. The proof proceeds by first showing that κ‑Frechet–Urysohn spaces satisfy the (α4) property, which allows one to extract a Mackey null subsequence from any null sequence. Then, using the equivalence of (K) and (MK) in spaces with (α4) (Proposition 2.5), the null sequence becomes a K‑sequence, and the classical Baire argument for metrizable (K) spaces extends to this non‑metrizable setting.

A direct corollary (Corollary 1.11) states that every locally complete κ‑Frechet–Urysohn lcs is Baire. This simultaneously generalizes two earlier theorems: (i) Burzyk, Kliś and Lipecki’s result that metrizable (K) spaces are Baire, and (ii) Kąkol and Sánchez Ruiz’s theorem that sequentially complete Frechet–Urysohn lcs are Baire.

The paper also explores the independence of the involved properties. It constructs a “feral” Baire space E (all bounded subsets are finite‑dimensional) that satisfies (K) but fails to be κ‑Frechet–Urysohn, showing that (K) does not imply κ‑Frechet–Urysohn. Conversely, Example 4.1 exhibits a κ‑Frechet–Urysohn lcs that is not Baire, demonstrating that κ‑Frechet–Urysohn alone is insufficient. Moreover, Example 4.7 provides a Baire lcs that is feral and not κ‑Frechet–Urysohn, answering negatively the question “Does Baire ⇒ κ‑Frechet–Urysohn for lcs?”

Despite the failure of full Baire-ness, the authors prove that every κ‑Frechet–Urysohn lcs is b‑Baire‑like in the sense of Ruess (Theorem 5.1). This property implies quasibarrelledness and yields many new examples of b‑Baire‑like spaces, including several Cₖ‑spaces. Theorem 5.2 gives a precise characterization of κ‑Frechet–Urysohn locally convex spaces that are Baire‑like.

In the final section, the authors apply their main theorem to function spaces. They give concise proofs that for a k‑space X, the space Cₖ(X) of continuous real‑valued functions with the compact‑open topology is Baire if and only if it is κ‑Frechet–Urysohn, which in turn is equivalent to X possessing Sakai’s (κκ) property. Similar equivalences are obtained for spaces of Baire‑α functions B_α(X). These results recover and strengthen known theorems about Baire functions and Cₖ‑spaces, but now follow directly from the general (MK) framework.

Overall, the paper introduces property (MK) as a natural bridge between Mackey null sequences and K‑sequences, shows that local completeness guarantees (MK), and demonstrates that κ‑Frechet–Urysohn together with (MK) suffices for Baire-ness. By providing counterexamples, establishing b‑Baire‑like behavior, and delivering applications to classical function spaces, the work significantly broadens the understanding of Baire phenomena in topological vector spaces.