Finite and Countable Additivity of Topological Properties in Nice Spaces

Abstract

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