Espaces non vérifiant le second axiome de dénombrabilité. (Spaces which do not satisfy the second axiom of countability)
Let E be a topological space and \(\tau\) : \({\mathcal P}(E)\to [0,\infty]\) be an increasing function such that \(\tau (\emptyset)=0\). A set \(A\subset E\) is said to be \(\tau\)-compact iff for any open covering \({\mathcal G}\) of A and for every \(\epsilon >0\) there is a finite subfamily \({\mathcal G}_ 0\subset {\mathcal G}\) such that \(\tau\) (A\(\setminus \cup {\mathcal G}_ 0)<\epsilon\). This notion has some ``good'' properties. For instance, \(\tau\)-compactness is hereditary with respect to closed subsets, it admits also the following separate property: if E is separable, \(A\subset E\) is \(\tau\)-compact, and \(x\in E\setminus A\), then for every \(\epsilon >0\) there exists a neighbourhood V of x such that \(\tau (A\cap V)<\epsilon.\) It is proved that if \(\tau\) is finitely subadditive, E is \(\tau\)-compact, \(A\subset E\), and \(\tau (A)>0\), then there exists a point of E which is adherent to A, i.e. a point \(x\in E\) such that \(\tau (A\cap V)>0\) for every neighbourhood V of x. Another result states that if \(\tau\) is an outer measure and there exists a set of positive outer measure \(\tau\) having no \(\tau\)-adherent point, then E does not satisfy the second axiom of countability and \(\tau\) is weakly Archimedean, i.e. \(\lim_{\epsilon \to 0}\tau (\cup \{A\subset E: \tau (A)<\epsilon \})>0.\) Some other theorems in this spirit use also the concept of the so-called black holes. One of them gives another condition sufficient for an outer measures to be weakly Archimedean.
scientific article; zbMATH DE number 4055965