Semiregular properties and generalized Lindelöf spaces

Abstract

Let $(X,\tau )$ be a topological space and $(X,{\tau }^{\ast })$ its semiregularization. Then a topological property $\mathcal{P}$ is semiregular provided that $\tau$ has property $\mathcal{P}$ if and only if ${\tau }^{\ast }$ has the same property. In this work we study semiregular property of almost Lindelöf, weakly Lindelöf, nearly regular-Lindelöf, almost regular-Lindelöf and weakly regular-Lindelöf spaces. We prove that all these topological properties, on the contrary of Lindelöf property, are semiregular properties.