Neeran Tahir Abd Al Ameer Al Khafaji

The main aim of this paper to study new class of continuous function is called bc-continuous function, For this aim , the nation of b-open and α-open sets and b-compact space introduced, and we shall study the relationship between bc-continuous and αc-continuous .

In this paper we obtain a formula for the order of the In semigroup (The symmetric Inverse Semigroup) and a formula to find the number of idempotent elements in it also we prove that this number always even.

In this paper we introduce the concept of fuzzy semi - T* 1\2 space and we study their relation ship with the

In this paper we obtain a formula for the order of elements in Tn (The symmetric Semigroup) and Pn(The Partial symmetric Semigroup ) semigroups .

In this paper, the authors studies new types of sets called β -sets in topological space (X. τ). We investigate and prove some topological properties and characterization of β-sets. Also, the relation between β- sets and open sets is discussed.

Bousfield introduced an algebraic category of modules that reflects the structure detected by p-localized complex topological K-theory. He constructed, for any module M in this category, a natural 4-term exact sequence , where U denotes the co-free functor, right adjoint to the forgetful functor to -modules. Clarke et al. identified the objects of Bousfield’s category as the ‘discrete’ modules for a certain topological ring A, obtained as a completion of the polynomial ring , and simplified the construction of the Bousfield sequence in this context. We introduce the notion of ‘cobasically discrete’ R-modules as a clarification of the Clarke et al. modules, noting that these correspond to comodules over the coalgebra that R is dual to. We study analogues of the Bousfield sequence for other polynomial completion rings, noting a variety of behaviour in the last term of the sequence.