On \(\theta\)-connectedness and \(\theta\)-closure spaces (Q697604)

If \((X,{\mathcal T})\) is a topological space and \(A\subseteq X\), then the \(\theta\)-closure of \(A\), denoted by cl\(_\theta(A)\) is the set \(\{x:\) each closed neighbourhood of \(x\) meets \(A\}\). A set \(C\) is \(\theta\)-closed if \(C=\text{ cl}_\theta(C)\); the \(\theta\)-closed sets form the closed sets of a topology called the \(\theta\)-topology, denoted here by \({\mathcal T}_\theta\). The \(\theta\)-closure operator is not idempotent but is a Čech closure operator. This paper studies properties of the \(\theta\)-topology and the semiregularization topology \({\mathcal T}_s\) (the topology generated by the regular open sets). Among the results it is shown that local connectedness (respectively, compactness, normality) of \((X,{\mathcal T})\) implies the corresponding condition for \((X,{\mathcal T}_s)\) which in turn implies that \((X,{\mathcal T}_\theta)\) is locally connected (respectively, compact, normal).