This note establishes that the familiar internal characterizations of the Tychonoff spaces whose rings of continuous real-valued functions are complete, or -complete, as lattice ordered rings already hold in the larger setting of pointfree topology. In addition, we prove the corresponding results for rings of integer-valued functions.
Modifying Bowen's entropy, we introduce a new uniform entropy. We prove that the completion theorem for uniform entropy holds in the class of all metric spaces. However, the completion theorem for Bowen's entropy does not hold in the class of all totally bounded metric spaces.
We study the classes of complex Banach spaces with Valdivia dual unit ball. We give complex analogues of several theorems on real spaces. Further we study relationship of these complex Banach spaces with their real versions and that of real Banach spaces and their complexification. We also formulate several open problems.
Nous donnons, pour chaque niveau de complexité Γ, une caractérisation du type "test d'Hurewicz" des boréliens d'un produit de deux espaces polonais ayant toutes leurs coupes dénombrables ne pouvant pas être rendus Γ par changement des deux topologies polonaises.
We show that each of the classes of hereditarily locally connected, finitely Suslinian, and Suslinian continua is Π₁¹-complete, while the class of regular continua is Π₀⁴-complete.