Baire-type properties of topological vector spaces
Saak Gabriyelyan, Alexander Osipov, Evgenii Reznichenko
TL;DR
The paper investigates when topological vector spaces exhibit the Baire property under weaker assumptions than classical completeness or $(K)$. It introduces the Mackey-null based property $(MK)$ and proves that $(K)\Rightarrow (MK)$, with $(MK)$ sufficing for $\kappa$-Fréchet--Urysohn spaces to be Baire; locally complete $\kappa$-FU lcs are thus Baire, generalizing prior results. It also demonstrates independence among $(MK)$, $\kappa$-FU, and Baireness via feral and non-$\kappa$-FU examples, and derives significant consequences for spaces of Baire and continuous functions, including $C_k(X)$ and $B_\alpha(X)$. The findings clarify the landscape of Baire-type properties in topological vector spaces and yield new, streamlined proofs for classical results in function spaces, with implications for $C_k(X)$ and related spaces.
Abstract
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.