This paper is devoted to introduce new concepts so called K( c)-spaces several various theorems about these concepts are provided in addition, further properties are studied such as the relationships between those concepts and other types of K( c)-spaces are investigated.

Abstract. This paper is devoted to introduce new concepts so called m-L(sc)-spaces. Several theorems related to these concepts are proved, further properties are studied as well as the relationships between these concepts with another types of m-L(sc)-spaces are investigated. Key words.mx-open set, -compact, -lindelof , -L -spaces, -lindelof and mx -semi closed. IntroductionIt is known that there is no relation between -Lindelof space and mx-closed sets, so this point stimulated some researchers to introduce a new concept namely -Lc-spaces [1], these are the spaces ( -Lc-spaces) in which every -Lindelof subset is mx-closed. In 2015 the author [2] introduced a new concept, namely, -2 (=A non-empty set with an m-space is said to be -2 if mx-cl ( ) is m-compact in for a subset of a -space , whenever iscompact). The basic definitions that are needed in this work are recalled. A space( ,mx) means a mspace where a sub family mx of the power set ( ) , such that and belong to mx [3]. Each member of mx is said to be mx-open set and the complement of an mx-open set is said to be mxclosed set. We denote the ( ,mx) by m-space. For a subset of a m-space , the mx-interior of and the mx-closure of are defined as follows : mx-( ) =∩{ : ⊆ , -is mx-open} mx-( ) = ⋃{ : ⊆ , ∈ } Note that mx− ( )(mx− The m-discrete space ( , ), where is infinite countable set, and = -Space, then ( , ) is -lindelof, which is not -compact [2]. An m-space which has ( ) property is mx-T1-(

We wrote this paper to proffer new types of the perfectly supra continuous functions. We also introduced new types of supra continuous, supra open and supra closed functions.

This paper is devoted to introduce new concepts so called K(c)-space several various theorems about these concepts are provided In addition, further properties are studied such as the relationships between those concepts and other types of K(c)-spaces are investigated.

The aim of this paper is to add new types of continuous functions and results in the specialization field, where we merge two important terms of open sets that are pre-open and ω-open to get a new set named ωp-open set, where we introduce new functions by using this set, such as Mωpc, ωpMc, ωpMωpc, ωp-continuous, ωp*-continuous, and ωp-irresolute function. We clarify the relationship between these types and illustrate their relationship with some other types of continuous functions. Also we define some new types of spaces such as, ωp-compact, ωp-Lindelöf, and Cωp-Lindelöf. In addition, we submit important results like, the ωp*-continuous image of compact space being ωp-compact, and if X is a ωp-regular space, hence it is Cωp-Lindelöf, as well as we provide the composition between most of the new functions that we defined. Many other results have been found in this our work and they are supported by many examples.