Some results on quasi-\(\sigma\) and \(\theta\) spaces (Q2747323)

Following \textit{R. E. Hodel} [Duke Math. J. 39, 253-263 (1972; Zbl 0242.54027)] a topological space \((X,\tau)\) is called a \(\theta\)-space if there is a function \(g:\omega\times X\to\tau\) having the following properties: (i) \(x\in\bigcap \{g(n,x) \mid n<\omega\}\) for each \(x\) in \(X\); (ii) \(g(n+1,x) \subset g(n,x)\) for all \(n<\omega\) and \(x\) in \(X\); (iii) if for each \(n<\omega\), \(\{x,x_n\} \subset g(n,y_n)\) and \(y_n\in g(n,x)\), then \(x\) is a cluster point of the sequence \((x_n)\). If condition (iii) is weakened by just requiring that \((x_n)\) has some cluster point, then \((X,\tau)\) is called a \(w\theta\)-space. These spaces have been studied intensively by \textit{P. Fletcher} and \textit{W. F. Lindgren} [General Topology Appl. 9, 139-153 (1978; Zbl 0394.54015)]. In the present paper the following interesting facts concerning \(\theta\)-spaces and \(w\theta\)-spaces are proved.NEWLINENEWLINENEWLINETheorem 1: Every linearly ordered topological space with a quasi-\(G_\delta\)-diagonal is a \(\theta\)-space. NEWLINENEWLINENEWLINETheorem 2: Every \(w\theta\)-space with a quasi-\(G^*_\delta\)-diagonal is a \(\theta\)-space.NEWLINENEWLINENEWLINETheorem 3: A topological space with a quasi-\(G^*_\delta\)(2)-diagonal is a \(\Theta\)-space in the sense of P. Fletcher if and only if it is a \(w\theta\)-space.NEWLINENEWLINENEWLINETheorem 4: Every linearly ordered topological space with a quasi-\(G^*_\delta(2)\)-diagonal is a \(\Theta\)-space.NEWLINENEWLINENEWLINETheorem 5: A \(w\theta\)-space is metrizable if and only if it is a quasi-Nagata space with a quasi-\(G^*_\delta (2)\)-diagonal.NEWLINENEWLINENEWLINETheorem 6: A topological space is developable if and only if it is a \(w\theta\), \(\beta\)-space with a quasi-\(G^*_\delta(2)\)-diagonal.NEWLINENEWLINENEWLINEAdditionally, the paper contains two results concerning quasi-\(\sigma\)-spaces.