A new class of fuzzy closed sets, namely fuzzy weakly closed set in a fuzzy topological space is introduced and it is established that this class of fuzzy closed sets lies between fuzzy closed sets and fuzzy generalized closed sets. Alongwith the study of fundamental results of such closed sets, we define and characterize fuzzy weakly compact space and fuzzy weakly closed space.

Machines whose main purpose is to permute and sort data are studied. The sets of permutations that can arise are analysed by means of finite automata and avoided pattern techniques. Conditions are given for these sets to be enumerated by rational generating functions. As a consequence we give the first non-trivial examples of pattern closed sets of permutations all of whose closed subclasses ha...

The members of Martin-Löf random closed sets under a distribution studied by Barmpalias et al. are exactly the infinite paths through Martin-Löf random Galton-Watson trees with survival parameter 2 3 . To be such a member, a sufficient condition is to have effective Hausdorff dimension strictly greater than γ = log2 3 2 , and a necessary condition is to have effective Hausdorff dimension greate...

We study Michael’s lower semifinite topology and Fell’s topology on the collection of all closed limit subsets of a topological space. Special attention is given to the subfamily of all maximal limit sets.

An effectively closed set, or Π 1 class, may viewed as the set of infinite paths through a computable tree. A numbering, or enumeration, is a map from ω onto a countable collection of objects. One numbering is reducible to another if equality holds after the second is composed with a computable function. Many commonly used numberings of Π 1 classes are shown to be mutually reducible via a compu...