Regular \(L\)-fuzzy topological spaces and their topological modifications
Let \(L\) be a continuous lattice with its Scott topology. There is a functor \(\omega_L\) from the category \textbf{Top} of topological spaces to the category \(L\)-\textbf{Top} of \(L\)-topological spaces. \(\omega_L\) sends every topological space \((X,{\mathcal T})\) to \((X,\omega_L({\mathcal T}))\), where \(\omega_L ({\mathcal T})\) denotes the collection of all continuous functions from \(X\) to \(L\). Conversely, there is also a functor \(\iota_L\) from \(L\)-\textbf{Top} to \textbf{Top}, \(\iota_L\) sends every \(L\)-topological space \((X,\Delta)\) to \((X,\iota_L (\Delta))\), where \(\iota_L(\Delta)\) is the coarsest topology on \(X\) making all \(\lambda \in \Delta\) continuous. In this paper it is showed that both of the functors \(\omega_L\) and \(\iota_L\) preserve regularity. Precisely, if \((X,{\mathcal T})\) is regular then \((X,\omega_L ({\mathcal T}))\) is \(L\)-regular (i.e., regular in the Hutton-Reilly sense); if \((X,\Delta)\) is \(L\)-regular then \((X,\iota_L (\Delta))\) is regular. Moreover, the relationship of \(L\)-regular spaces to \(H\)-Lindelöf spaces and \(L\)-normal spaces is discussed.
scientific article; zbMATH DE number 1496729