Fuzzy topologies with specified level properties

Abstract Recently the author and P. Wuyts have studied the problem of defining fuzzy topologies when one is given the family of α-closure operators or the family of level topologies. This paper is concerned with the relationship between these two problems. When the family of topologies is a chain with the LT-property, there is a natural family of closure operators that generates the Wuyts maximum, i.e., the largest fuzzy topology with the given level topologies. As a corollary, it is shown that the fuzzy neighborhood spaces are precisely the α-pseudo-regular spaces. In general, the relationship between the two problems is weak. For example, given an LT-family of topologies and an L-FTP family of semi-closure operators, there is a fuzzy topology associated with both if and only if the 0-level closure generates the 0-level topology and the α-closure of a set is always closed in the α-level topology. Lastly, it is shown that, when a function's domain has a Wuyts maximum as its topology, fuzzy continuity is equivalent to level continuity.