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Insertion of lattice-valued functions in a monotone manner is investigated. For L a -separable completely distributive lattice (i.e. L admits a countable base which is free of supercompact elements), a monotone version of the Kat¥tovTong insertion theorem for L-valued functions is established. We also provide a monotone lattice-valued version of Urysohn's lemma. Both results yield new characterizations of monotonically normal spaces. Moreover, extension of lattice-valued functions under additional assumptions is shown to characterize also monotone normality.

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Abstract. For L a continuous lattice with its Scott topology, the functor ι L makes every regular L-topological space into a regular space and so does the functor ω L the other way around. This has previously been known to hold in the restrictive class of the socalled weakly induced spaces. The concepts of H-Lindelöfness (á la Hutton compactness) is introduced and characterized in terms of certain filters. Regular H-Lindelöf spaces are shown to be normal.Keywords and phrases. Fuzzy topology, regularity, the functors ι L and ω L , H-Lindelöf property.2000 Mathematics Subject Classification. Primary 54A40. In this paper, we show that when L is a continuous lattice with its Scott topology then ι L maps the category Reg(L) of L-regular spaces onto the category Reg of regular spaces. This improves upon and extends a result of Liu and Luo [6] which showed (with different but equivalent terminology) that ι L maps weakly induced L-regular spaces to regular spaces (with L a completely distributive lattice with its upper topology). As a consequence, we have that ω L (Reg) consists precisely of L-regular spaces of ω L (TOP). Some generalities about L-regular spaces are included and stated in a slightly more general situation, viz. for L-topologies that admit a certain type of approximating relation. This captures complete L-regularity and zero-dimensionality. Introduction. The two functors that provide a working link between the category TOP(L) of L-(fuzzy)-topologicalWe also introduce the concept of H-Lindelöfness (compatible with compactness in the sense of Hutton [2]) and characterize it in terms of closed filters. Finally, we prove that H-Lindelöf and L-regular spaces are L-normal.

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