Finite and countable additivity of topological properties in nice spaces

Abstract

Let Q ∈ Q \in character ≤ τ \leq \tau , pseudocharacter ≤ τ \leq \tau , tightness ≤ τ \leq \tau , weight ≤ τ \leq \tau , P τ {P_\tau } -property, discreteness, Fréchet-Urysohn property, sequentiality, radiality, pseudoradiality, local compactness, k k -property. If X n = ∪ { X i : i ∈ n } {X^n} = \cup \{ {X_i}:i \in n\} , X i ⊢ Q {X_i} \vdash Q for all i ∈ n i \in n then X ⊢ Q X \vdash Q (i.e. the property Q Q is n n -additive in X n {X^n} for any X ∈ T 3 X \in {T_3} ). Metrizability is n n -additive in X n {X^n} provided X X is compact or c ( X ) = ω c(X) = \omega . ANR {\text {ANR}} -property is closely n n -additive in X n {X^n} if X X is compact ("closely" means additivity in case X i {X_i} is closed in X n {X^n} ). If Q ∈ Q \in metrizability, character ≤ τ \leq \tau , pseudocharacter ≤ τ \leq \tau , diagonal number ≤ τ \leq \tau , i i -weight ≤ τ \leq \tau , pseudoweight ≤ τ \leq \tau , local compactness then Q Q is finitely additive in any topological group.