Semiregular properties and generalized Lindelöf spaces

Abstract

Let $(X, \tau)$ be a topological space and $(X,\tau^{\ast})$ its semiregularization. Then a topological property ${\Cal P}$ is semiregular provided that $\tau$ has property ${\Cal P}$ 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.