Separations in lattice-valued induced spaces

Abstract

For the value set (i.e. range) I, the induced space is normal if and only if its underlying space is normal. But for a general value set, the situation is quite different. In this paper, we construct a skillful example to show that the corresponding relation about the normalities of these two spaces does not hold for a certain fuzzy lattice. Meanwhile, for an important type of fuzzy lattices, using the new method, we successfully establish a positive result about this normality relation. Furthermore, we shall describe an α-level set of the closure (or interior) of a fuzzy set A in terms of the topology of the underlying space and some level sets of A. This construction problem of closed sets and open sets in fuzzy topological spaces possesses independent interest. Other separation axioms, such as Hausdorff separation and regularity, are also discussed in this paper.