Relations between new topologies obtained from old ones (Q1333051)

This paper considers relationships between several topologies defined on the same underlying set. If \((X, \tau)\) is a topological space then the topology \(\tau_s\) is the semi-regularization of \(\tau\), \(\tau_\theta\) is the topology of \(\theta\)-open subsets of \((X, \tau)\) and \(\tau^\alpha\) is the topology of \(\alpha\)-open subsets of \(X\). A family \({\mathcal A}\) of subsets of \(X\) is defined to be a supratopology on \(X\) if \(\emptyset \in {\mathcal A}\), \(X \in {\mathcal A}\) and \({\mathcal A}\) is closed under arbitrary unions. The families of preopen subsets of \(X\), semi-open subsets of \(X\), and semi-preopen subsets of \(X\) are supratopologies on \(X\). If \({\mathcal A}\) is a supratopology on \(X\), then \(\tau_{\mathcal A} = \{T \subset X : A \in {\mathcal A}\) implies \(T \cap A \in {\mathcal A}\}\) is a topology on \(X\), and \(\tau_{\mathcal A} \subset {\mathcal A}\). The author uses this result to provide various relationships between these topologies, especially in the case when \((X, \tau)\) is extremally disconnected.