Some fundamental properties of maximal open sets are obtained, such as decomposition theorem for a maximal open set. Basic properties of intersections of maximal open sets are established, such as the law of radical closure. 1. Introduction. A proper nonempty open subset U of a topological space X is said to be a maximal open set if any open set which contains U is X or U. In [2], we study mini...

We prove that the number of minimal transversals (and also the number of maximal independent sets) in a 3-uniform hypergraph with n vertices is at most c, where c ≈ 1.6702. The best known lower bound for this number, due to Tomescu, is ad, where d = 10 1 5 ≈ 1.5849 and a is a constant.

Dynkin’s classification of maximal subalgebras of simple finite dimensional complex Lie algebras is generalized to matrix Lie superalgebras, i.e., the Lie subsuperalgebras of gl(p|q).

The trace of a family of sets A on a set X is A|X = {A ∩ X : A ∈ A}. If A is a family of k-sets from an n-set such that for any r-subset X the trace A|X does not contain a maximal chain, then how large can A be? Patkós conjectured that, for n sufficiently large, the size of A is at most ( n−k+r−1 r−1 ) . Our aim in this paper is to prove this conjecture.

The trace of a set F on a another set X is F |X = F ∩ X and the trace of a family F of sets on X is FX = {F |X : F ∈ F}. In this note we prove that if a k-uniform family F ⊂ ([n] k ) has the property that for any k-subset X the trace F|X does not contain a maximal chain (a family C0 ⊂ C1 ⊂ ... ⊂ Ck with |Ci| = i), then |F| ≤ ( n−1 k−1 ) . This bound is sharp as shown by {F ∈ ([n] k ) , 1 ∈ F}. ...

A nonzero fuzzy open set () of a fuzzy topological space is said to be fuzzy minimal open (resp. fuzzy maximal open) set if any fuzzy open set which is contained (resp. contains) in is either or itself (resp. either or itself). In this note, a new class of sets called fuzzy minimal open sets and fuzzy maximal open sets in fuzzy topological spaces are introduced and studied which are subclasses ...

The five simple exceptional complex Lie superalgebras of vector fields are described. One of them, kas, is new; the other four are explicitly described for the first time. All nonisomorphic maximal subalgebras of finite codimension of these Lie superalgebras, i.e., all other realizations of these Lie superalgebras as Lie superalgebras of vector fields, are also described; there are 14 of them a...

We show that in the one-dimensional case the weighted Hardy–Littlewood maximal operator Mμ is bounded on BMO(μ) for arbitrary Radon measure μ, and that this is not the case in higher dimensions.

In this paper we consider Q the class of solvable Lie algebras L with the following property: if A is a subalgebra of L, then Φ(A) ⊆ Φ(L) (where Φ(L) denotes the Frattini subalgebra of L;that is Φ(L) is the intersection of all maximal subalgebras of L). The class Q is shown to contain all solvable Lie algebras whose derived algebra is nilpotent. Necessary conditions are found such that an ideal...