Higher separation axioms in Z-topologically generated (L)-topological spaces

摘要

The main result of this paper is a theorem about inserting a pair of semicontinuous L-real-valued functions which extends the insertion theorem of Kubiak [Comment. Math. Univ. Carolinae 34 (1993) 357-362] from L = {0,1} to an arbitrary meet-continuous lattice L (endowed with an order-reversing involution). With this result it is shown that the normality-type separation axioms in TOP(L) are preserved by the functor which takes an L-topological space X to the (L)-topological space Ω_L(X) obtained by providing the set X with the (L)-topology consisting of all lower semicontinuous functions from X to (L). The same is proved for the case of the regularity axiom.