The number of \(K\)-determination of topological spaces
The paper is a study of two cardinal invariants: the number of \(K\)-determination \(\ell\Sigma\) and the number of Nagami \(Nag\) which are extensions of Lindelöf \(\Sigma\)-spaces and Nagami's spaces to higher cardinals. For a space \(Y\), \(\ell\Sigma(Y)\) (\(Nag(Y)\)) is the smallest infinite cardinal number \(\mathfrak{m}\) for which there are a metric space \(X\) (a space \(X\)) of weight \(\mathfrak{m}\) and an upper semicontinuous compact-valued mapping \(\phi:X \to Y\) such that \(Y = \bigcup\{\phi(x):x\in X\}\). The behaviour of these functions under several topological constructions is investigated, as well as their relationships with other cardinal invariants. For a topological space \(Y\) we have \(\ell(Y) \leq Nag(Y) \leq \ell\Sigma(Y) \leq |Y|\). If \(Y\) is a completely regular space, then \(Nag(Y) = \max\{\Sigma(Y),\ell(Y)\}\), where \(\Sigma(Y)\) is the \(\Sigma\)-degree of \(Y\). There is a space \(Y \subset \beta\mathbb N\) such that \(Nag(Y) \leq w(Y) < \ell\Sigma(Y)\). If \(Y\) is a compact space, then \(\ell\Sigma(C_p(Y)) \leq w(C_p(Y))\).
scientific article; zbMATH DE number 6132871
scientific article