The degree of weakly discretely generated spaces
A space \(X\) is discretely generated if for every subset \(A\) of \(X\) and every \(x\in \text{cl}\, A\), there is a discrete subset \(D\) of \(A\) such that \(x\in\text{cl}\, D\). \(X\) is weakly discretely generated if for every non-closed subset \(A\) of \(X\) there is a discrete subset \(D\) of \(A\) such that \(\text{cl}\, D\setminus A\neq\emptyset\). For every subset \(A\) of \(X\) and every ordinal \(\alpha\), \([A]_0=A\), \([A]_\alpha=\bigcup\{[A]_\beta:\beta< \alpha\}\) if \(\alpha\) is a limit ordinal and \([A]_\alpha= \bigcup \{\text{cl}\, D: D\) a discrete subset of \([A]_\gamma\}\) if \(\alpha= \gamma +1\). Finally, the degree of discrete generation of a weakly discretely generated space \(X\) is defined by \[ idc(X) =\min \{\alpha : [A]_\alpha = \text{cl}\, A \text{ for all subsets } A \text{ of } X\}. \] This paper contains several new results concerning these notions, particularly, in the classes of pseudocompact, locally compact and Čech-complete spaces.
scientific article; zbMATH DE number 6330522
scientific article