1. When a system of sets covers a space, the star number of the covering is the supremum over the sets of the system of the cardinals of the numbers of sets of the system meeting a set of the system . The standard Lebesgue 'brick-laying' construction provides an example, for each positive integer n, of a lattice covering of R" by closed rectangular parallelepipeds with star number 2"T'-1 . In v...

Given a nonempty set X, a fuzzy ideal I on X is a collection of subsets of X closed under finite union and subset operations. In this paper we define a space to be locally I fuzzy compact space if every point in the space has an I fuzzy compact neighbourhood. Basic results concerning locally I fuzzy compact spaces are given relating to subspaces, preservation of functions, and products. Classic...

We prove that it is relatively consistent with $\mathrm{ZFC}$ every strong measure zero subset of the real line meager-additive while there are uncountable sets (i.e., Borel's conjecture fails). This answers a long-standing question due to Bartoszynski and Judah.

A singular hyperbolic set is a partially hyperbolic set with singularities (all hyper-bolic) and volume expanding central direction [MPP1]. We study connected, singular-hyperbolic, attracting sets with dense closed orbits and only one singularity. These sets are shown to be transitive for most C r flows in the Baire's second category sence. In general these sets are shown to be either transitiv...

In this paper, direct proofs of some properties of new classes of fuzzy sets called fuzzy strongly g∗-closed sets and fuzzy g∗∗-closed sets are introduced. Examples are presented showing that some generalizations can not be obtained. MSC: 54A40 • 54A05