Let [n]:={1,2,…,n} and let a k-set denote set of cardinality k. A family sets is union-closed (UC) if the union any two in also family. Frankl’s conjecture states that for nonempty UC F⊆2[n] such F≠{∅}, there exists an element i∈[n] contained at least half F, where 2[n] denotes power on [n]. The 3-sets Morris smallest number distinct (whose n-set) ensure satisfied (in contains them ⌊n/2⌋+1 all ...

This study introduce a new type of closed sets in topology called Generalized h-closed (briefly, gh-closed) define as follow: E ⊆ χ be gh-closed set if CLh (E)⊆ U whenever and is open (χ,τ). The relation between other classes ( h-closed, g-closed, g -closed, g-closed αg-closed) are studied. Also, the notion gh-continuous mapping on topological space some properties proved. Finally, separation a...

In this paper we have introduced a new class of closed in topological spaces called g**-closed sets and study some its properties . Further introduce the concept g**-continuous functions , g**-irresolute As an application two news paces namely T **-space, **T - space .Further, g**-continuous, mappings are also investigated

The aim of this paper is to introduce the concept of generalized regular closed sets and study some of its properties . The corresponding topological space formed by the family of these sets is also studied. It may be noted that regular closed set doesn’t forms even a supra topological space. Various researchers had studied the concept of generalized closed sets and regular generalized closed s...

In this paper, we study the class of generalized b-closed sets and use this notion to consider new weak and stronger forms of continuities associated with these sets. We apply these notions to give new characterization of extremally disconnected spaces and also Tgs-spaces. 2000 Mathematics Subject Classification: 54A10, 54C8, 54C10, 54D10

A dcpo P is continuous if and only if the lattice C(P ) of all Scottclosed subsets of P is completely distributive. However, in the case where P is a non-continuous dcpo, little is known about the order structure of C(P ). In this paper, we study the order-theoretic properties of C(P ) for general dcpo’s P . The main results are: (i) every C(P ) is C-continuous; (ii) a complete lattice L is iso...

A family of sets is union-closed if it contains the union of any two of its elements. Some years ago, Reimer gave a lower bound for the average size of an element of a union-closed family consisting of m sets and, two years ago, Czédli did the same under the additional condition that our sets are contained in a set with n elements. Recently Tom Eccles and I have determined the minimum average s...

In this paper, another generalized class of τ called mildly g-ωclosed sets is introduced and the notion of mildly g-ω-open sets in topological spaces is introduced and studied. The relationships of mildly g-ω-closed sets with various other sets are investigated.

We investigate notions of randomness in the space C[2N] of nonempty closed subsets of {0, 1}N. A probability measure is given and a version of the Martin-Löf test for randomness is defined. Π2 random closed sets exist but there are no random Π1 closed sets. It is shown that any random closed set is perfect, has measure 0, and has box dimension log2 4 3 . A random closed set has no n-c.e. elemen...