2025/07/15 by Saak Gabriyelyan, Gabriyelyan, Saak, Evgenii Reznichenko +1
Mathematics · #54A05 #54B05 #54C35 #54D50 #Advanced Banach Space Theory #Advanced Topology and Set Theory #FOS: Mathematics #General Topology (math.GN) #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.2507.11220
openalex publication_date 2025/07/15 · openalex created_date 2025/10/08 · openalex updated_date 2026/07/28
We say that a Tychonoff space X is a κ-space if it is homeomorphic to a closed subspace of Cp(Y) for some locally compact space Y. The class of κ-spaces is strictly between the class of Dieudonné complete spaces and the class of μ-spaces. We show that the class of κ-spaces has nice stability properties, that allows us to define the κ-completion κX of X as the smallest κ-space in the Stone--Čech compactification βX of X containing X. For a point z∈βX, we show that (1) if z∈υ X, then the Dirac measure δz at z is bounded on each compact subset of Cp(X), (2) z∈ κX iff δz is continuous on each compact subset of Cp(X) iff δz is continuous on each compact subset of Cpb(X), (3) z∈υ X iff δz is bounded on each compact subset of Cpb(X). It is proved that κX is the largest subspace Y of βX containing X for which Cp(Y) and Cp(X) have the same compact subsets, this result essentially generalizes a known result of R.~Haydon.