Microsoft Word - 175 -187   175  Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (3) 2020       𝝎Mc –functions and 𝑵Mc-functions Abstract In this paper, we presented new types of Mc-function by using 𝜔-open and 𝑁-open sets some of them are weaker than Mc-function and some are stronger, which are 𝜔Mc-function, M𝜔c-function, 𝜔M𝜔c-function, 𝑁Mc-function, M𝑁c-function and 𝑁M𝑁c-function, also we submitted new kinds of continuous functions and compact functions and we illustrated the relationships between these types. The purpose of this paper is to expand the study of Mc- function and to get results that we need to find the relationship with the types that have been introduced. Keywords: 𝜔Mc-function, M𝜔c-function, 𝜔M𝜔c-function, 𝑁Mc-function, 𝑁M𝑁c-function. 1. Introduction In (2011) Farhan, Ala’A. Mahmood [1]. Introduced the concept of new function which was named by Mc-function where he defined it as (A function 𝑓 from any topological space 𝑋, Ʈ into any topological space 𝑌, Ʈ was called Mc-function iff 𝑓 𝒲 was a closed set in 𝑋 for any compact set 𝒲 in 𝑌). Moreover, in (2015) the researchers in [2]. introduced new types of Mc-functions by using g-closed sets which were M-gc, gM-c and gM-gc functions, where every Mc-function was M-gc function and gMc function, The notion of 𝜔-open sets submitted at the first time by Hdeib in [3]. While the notion of 𝑁-open sets submitted by Al-Omari in [4]. Where they defined these sets as (Any set 𝒲 in a topological space 𝑋, Ʈ was called 𝜔-open (respectively. 𝑁-open) set, if for any element 𝑎 ∈ 𝒲 there was an open set ℬ containing a, with ℬ-𝒲 was countable (respectively. finite) set. 𝒲 was 𝜔-closed set (respectively. 𝑁-closed set). After that many researchers used these sets and introduced new concepts in different kinds of topological spaces. In this paper, we used these two sets to introduce new types of Mc-function, namely 𝜔Mc-function, M𝜔c-function, 𝜔M𝜔c-function, 𝑁Mc-function, M𝑁c-function and 𝑁M𝑁c-function and we illustrated the relation between them and their relation with Mc-function, also we connected between these functions and 𝑇 -space, 𝜔𝑇 -space, 𝑁𝑇 -space, Kc- space, and c-c space. We used the same sets to define new forms of continuous and compact functions such as strongly 𝜔-continuous, strongly 𝑁-continuous, 𝜔-irresolute, 𝜔∗-compact, 𝑁∗-compact, 𝜔∗∗-compact, 𝑁∗∗-compact function, and we presented some propositions, remarks and examples to support our work. Ibn Al Haitham Journal for Pure and Applied Science Journal homepage: http://jih.uobaghdad.edu.iq/index.php/j/index Doi: 10.30526/ 33.3.2484 Article history: Received 7 September 2019, Accepted 13 October 2019, Published in July 2020. Nadia Kadum Humadi Haider Jebur Ali Department of Mathematics, College of Science, Mustansiriyah University, Iraq. m.a.nadia313@gmail.com drhaiderjebur@uomustansiriyah.edu.iq   176  Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (3) 2020 𝟏-𝝎Mc-functions and 𝑵Mc-functions. We recall the following facts which we need in this our work, after that we will introduce the definitions of new types of Mc-function and illustrate the relation between these types. Definition (1.1): A space 𝑋, Ʈ is called:- 1- A compact space if any open cover for 𝑋 possesses a finite sub cover [5]. 2- A 𝜔-compact space if any 𝜔-open cover for 𝑋 possesses a finite sub cover [5]. 3- An 𝑁-compact space if any 𝑁-open cover for 𝑋 possesses a finite sub cover [6]. Remark (1.2): Every closed set is 𝜔-closed [7]. (respectively 𝑁-closed [6].) set. And every 𝑁-open set is 𝜔-open set [8]. Also every 𝑁-closed set is 𝜔-closed set [8]. As well as every 𝜔-compact (respectively 𝑁-compact) set is compact [5, 9]. Example (1.3): 1-In the excluded point topological space (𝑋, Ʈ ) where 𝑋 is a finite set, the set {𝑥 ∘ is 𝜔-closed and 𝑁-closed set but not closed set, in which Ʈ = 𝑈 ⊆ 𝑋, 𝑥∘ ∉ 𝑈 for some 𝑥∘ ∈ 𝑋 ⋃ 𝑋 . 2-The set of irrational number 𝒬 in the co-finite topological space (ℛ, Ʈ is 𝜔-open set but not 𝑁-open set. 3-The set of rational number 𝒬 in the co-finite topological space (ℛ, Ʈ is 𝜔-closed set but not 𝑁-closed set. 4-The included point topological space (𝒵, Ʈ where Ʈ = {𝑈 ⊆ 𝑋, where 𝑥∘ ∈ 𝑈, for some 𝑥° in 𝑋}⋃{∅} is a compact space but not 𝜔-compact. Definition (1.4): The space 𝑋, Ʈ is called:- 1- A 𝑇 -space, if for each non-equal elements 𝑥, 𝑦 in 𝑋, there are disjoint 𝒲 , 𝒲 ∈ Ʈ with 𝑥 ∈ 𝒲 and 𝑦 ∈ 𝒲 [10]. 2- A 𝜔𝑇 -space, if for each non-equal elements 𝑥, 𝑦 in 𝑋, there are disjoint 𝜔-open sets 𝒲 , 𝒲 in 𝑋 with 𝑥 ∈ 𝒲 and 𝑦 ∈ 𝒲 [11]. 3- An 𝑁𝑇 -space, if for each non-equal elements 𝑥, 𝑦 in 𝑋, there are disjoint 𝑁-open sets 𝒲 , 𝒲 with 𝑥 ∈ 𝒲 and 𝑦 ∈ 𝒲 [6]. Example (1.5) 1- ℛ, Ʈ is 𝑇 -space. 2- 𝑋, Ʈ where 𝑋 is a finite set and Ʈ = {𝒲 ⊆ 𝑋 | 𝒶∘ ∈ 𝒲, for some 𝒶∘ ∈ X ⋃ ∅ , is ωT -space and 𝑁T -space. Remark (1.6) 1- Every closed subset of a compact space is a compact set [6]. 2- Every 𝜔-closed subset of an 𝜔-compact space is an 𝜔-compact set [12]. 3- Every 𝑁-closed subset of an 𝑁-compact space is an 𝑁-compact set [6]. 4- Every compact subset of a 𝑇 -space is a closed set [12]. 5- Every 𝜔-compact subset of an 𝜔𝑇 -space is an 𝜔-closed set [12]. 6- Every 𝑁-compact subset of an 𝑁𝑇 -space is an 𝑁-closed set [13]. 7- Every 𝑇 -space is 𝜔𝑇 -space [14]. 8- Every 𝑇 -space is 𝑁𝑇 -space [6].   177  Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (3) 2020 Definition (1.7) A function 𝑓: 𝑋, Ʈ ⟶ 𝑌, Ʈ is called M𝜔c-function if 𝑓 𝒲 is a 𝜔-closed set in 𝑋 for any compact set 𝒲 in 𝑌. Definition (1.8) A function 𝑓: 𝑋, Ʈ ⟶ 𝑌, Ʈ is called M𝑁c-function if 𝑓 𝒲 is an 𝑁-closed set in 𝑋 for any compact set 𝒲 in 𝑌. Example (1.9) The identity function 𝐼 : 𝑋, Ʈ ⟶ 𝑋, Ʈ where 𝑋 is a denumerable set (infinite countable set), is M𝜔c-function. But in case 𝑋 is a finite set then 𝑓 is M𝑁c-function. Definition (1.10): A function 𝑓: 𝑋, Ʈ ⟶ 𝑌, Ʈ is called 𝜔Mc-function if 𝑓 𝒲 is a closed set in 𝑋 for any 𝜔-compact set 𝒲 in 𝑌. Definition (1.11) A function 𝑓: 𝑋, Ʈ ⟶ 𝑌, Ʈ is called 𝑁Mc-function if 𝑓 𝒲 is a closed set in 𝑋 for any 𝑁-compact set 𝒲 in 𝑌. Example (1.12) The identity function 𝐼ℛ : ℛ, Ʈ ⟶ ℛ, Ʈ is 𝜔Mc-function and 𝑁Mc-function. Definition (1.13) A function 𝑓: 𝑋, Ʈ ⟶ 𝑌, Ʈ is called 𝜔M𝜔c-function if 𝑓 𝒲 is an 𝜔-closed set in 𝑋 for any 𝜔-compact set 𝒲 in 𝑌. Example (1.14) The function 𝑓: 𝑁, Ʈ ⟶ ℛ, Ʈℛ is 𝜔M𝜔c-function. Definition (1.15) The function 𝑓: 𝑋, Ʈ ⟶ 𝑌, Ʈ is called 𝑁M𝑁c-function if 𝑓 𝒲 is an 𝑁-closed set in 𝑋 for any 𝑁-compact set 𝒲 in 𝑌. Example (1.16) The function 𝑓: 𝑋, Ʈ ⟶ ℛ, Ʈℛ where 𝑋 is a finite set, is 𝑁M𝑁c-function. The following scheme is helpful Mc-function M𝜔c-function (resp. M𝑁c-function) 𝜔Mc-function (resp. 𝑁Mc-function) 𝜔M𝜔c-function (resp. 𝑁M𝑁c- function) Remark (1.17) 1- Every 𝑁Mc-function is 𝜔Mc-function. 2- Every M𝑁c-function is M𝜔c-function. 3- Every 𝑁M𝑁c-function is 𝜔M𝜔c-function. Example (1.18) 1- 𝐼𝒵 : 𝒵, Ʈ ⟶ 𝒵, Ʈ is M𝜔c-function but neither M𝑁c-function nor Mc-function. 2- 𝐼 : 𝑋, Ʈ ⟶ 𝑋, Ʈ where 𝑋 is a finite set, is M𝑁c-function but not Mc-function.   178  Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (3) 2020 3- 𝑓: 𝒵, Ʈ ⟶ 𝒵, Ʈ is 𝜔M𝜔c-function but not 𝜔Mc-function. 4- 𝑓: 𝑋, Ʈ ⟶ 𝑋, Ʈ where 𝑋 is a finite set, is 𝑁M𝑁c-function but not 𝑁Mc-function. 5- 𝐼 : 𝑋, Ʈ ⟶ 𝑋, Ʈ where 𝑋= {1, 2, 3} and Ʈ= {∅, 𝑋, 1 } is 𝜔M𝜔c-function and 𝑁M𝑁c-function, but not Mc-function. 6- 𝐼𝒵 : 𝒵, Ʈ ⟶ 𝒵, Ʈ is M𝜔c-function, but not 𝜔Mc-function. 7- 𝐼 : 𝑋, Ʈ ⟶ 𝑋, Ʈ  where 𝑋 is a finite set, is M𝑁c-function but not 𝑁Mc-function. 8- 𝐼𝒵 : 𝒵, Ʈ ⟶ 𝒵, Ʈ is 𝜔Mc-function but not 𝑁Mc-function. 9- 𝐼𝒵 : 𝒵, Ʈ ⟶ 𝒵, Ʈ is M𝜔c-function and 𝜔M𝜔c-function but neither 𝑁M𝑁c- function nor M𝑁c-function. Definition (1.19) A function 𝑓: 𝑋, Ʈ ⟶ 𝑌, Ʈ is called:- 1- A continuous function If 𝑓 𝒲 is open set in 𝑋 for any open set 𝒲 in the space 𝑌 [15]. 2- An 𝜔-continuous function if 𝑓 𝒲 is 𝜔-open set in 𝑋 for each open set 𝒲 in 𝑌 [16]. 3- An 𝑁-continuous function if 𝑓 𝒲 is 𝑁-open set in 𝑋 for each open set 𝒲 in 𝑌 [15]. 4- A strongly 𝜔-continuous function if 𝑓 𝒲 is open set in 𝑋 for each 𝜔-open set 𝒲 in 𝑌. 5- A strongly 𝑁-continuous function if 𝑓 𝒲 is open set in 𝑋 for each 𝑁-open set 𝒲 in 𝑌. 6- An 𝜔‐irresolute function if 𝑓 𝒲 is 𝜔-open set in 𝑋 for each 𝜔-open set in 𝑌. 7- An 𝑁-irresolute function if 𝑓 𝒲 is 𝑁-open set in 𝑋 for each 𝑁-open set 𝒲 in 𝑌 [9]. Example (1.20) 1- 𝑓: 𝑋, Ʈ ⟶ 𝑌, Ʈ is continuous function. 2- The function 𝑓 from the excluded point space 𝑋, Ʈ where 𝑋 is a countable set into any space 𝑌, Ʈ is 𝜔-continuous function, where Ʈ 𝒲 ⊆ 𝑋, 𝑎∘ ∉ 𝒲 for some 𝑎∘ ∈ 𝑋 ⋃ 𝑋 . 3- The function 𝑓 from the included point space 𝑋, Ʈ into the discrete space 𝑋, Ʈ where 𝑋 is a finite set, is 𝑁-continuous function. 4- 𝑓: 𝑋, Ʈ ⟶ 𝑋, Ʈ , where 𝑓 𝑎 𝑑 for any 𝑎 ∈ 𝑋, is strongly 𝜔-continuous function. 5- The identity function from the co-countable space 𝑋, Ʈ into the same space where 𝑋 is uncountable set, is strongly 𝑁-continuous function. 6- A function 𝑓 from the included point space 𝒵, Ʈ into the co-finite space 𝒵, 𝜇 is 𝜔- irresolute function. 7- The function 𝑓 from elective topology on an infinite set to the indiscrete topology on the same set in which 𝑓 𝑎 𝑑 for any 𝑎 in the domain, is 𝑁-irresolute function. The following scheme is useful. continuous function 𝜔-continuous (resp. 𝑁-continuous) function strongly 𝜔-continuous (resp. strongly 𝜔-irresolute (resp. 𝑁-irresolute) function 𝑁- continuous) function Remark (1.21) 1- Every 𝑁-continuous function is 𝜔-continuous. 2- Every strongly 𝜔-continuous function is strongly 𝑁-continuous. 3- No relation between 𝜔-irresolute function and 𝑁-irresolute.   179  Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (3) 2020 Example (1.22) 1- 𝑓: 𝒵, Ʈ ⟶ 𝒵, Ʈ  is 𝜔-continuous but not 𝑁-continuous function. 2- 𝐼𝒵 : 𝒵, Ʈ ⟶ 𝒵, Ʈ   is strongly 𝑁-continuous but not strongly 𝜔-continuous function. 3- 𝐼𝒵 : 𝒵, Ʈ ⟶ 𝒵, Ʈ is 𝜔-irresolute but not 𝑁-irresolute function. Definition (1.23) 1- A space 𝑋, Ʈ) is called:- 1- c-c space if each closed set in 𝑋 is compact and each compact set is closed [17]. 2- Kc-space if each compact set in 𝑋 is closed [18]. Example (1.24) 1- 𝑋, Ʈ ) where 𝑋 is a finite set is c-c space. 2- ℛ, Ʈ𝓊 is Kc-space. Remark (1.25) Every 𝑇 -space is Kc-space. Example (1.26): (ℛ, Ʈ is Kc-space but not 𝑇 -space. Proposition (1.27) If 𝑓: 𝑋, Ʈ ⟶ 𝑌, Ʈ is a 𝜔-continuous function where 𝑌 is a 𝑇 -space, then it is an M𝜔c-function. Proof Let 𝒦 be a compact subset of 𝑌, since 𝑌 is a 𝑇 -space then 𝒦 is closed (by remark (1.6)), but 𝑓 is 𝜔-continuous function, so 𝑓 𝒦 is 𝜔-closed set in 𝑋, therefore 𝑓 is M𝜔c-function. By the same way we can prove the following proposition. Proposition (1.28) If 𝑓: 𝑋, Ʈ ⟶ 𝑌, Ʈ is:- 1- An 𝑁-continuous function where 𝑌 is a 𝑇 -space, then it is an M𝜔c-function (respectively M𝑁c-function). 2- An 𝜔-continuous function where 𝑌 is a Kc-space (respectively c-c space), then it is an M𝜔c-function. 3- An 𝑁-continuous function where 𝑌 is a Kc-space (respectively c-c space), then it is an M𝑁c-function. 4- A continuous function where 𝑌 is a 𝑇 -space (respectively Kc-space, or c-c space), then it is an M𝜔c-function, M𝑁c-function, 𝜔Mc-function,  𝑁Mc-function, 𝜔M𝜔c-function and 𝑁M𝑁c-function. 5- A strongly 𝜔-continuous function where 𝑌 is a 𝜔𝑇 -space (respectively Kc-space, c-c space), then it is a 𝜔Mc-function and 𝜔M𝜔c-function. 6- A strongly 𝜔-continuous function where 𝑌 is a 𝑁𝑇 -space (respectively Kc-space, c-c space), then it is an 𝑁Mc-function and 𝑁M𝑁c-function. 7- A strongly 𝑁-continuous function where 𝑌 is a 𝑁𝑇 -space, then it is an 𝑁Mc-function. 8- An 𝜔-irresolute function where 𝑌 is a 𝜔𝑇 -space (respectively Kc-space, c-c space), then it is an 𝜔M𝜔c-function. 9- An 𝑁-irresolute function where 𝑌 is a 𝑁𝑇 -space (respectively Kc-space, c-c space), then it is an 𝑁M𝑁c-function.   180  Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (3) 2020 Proof: 1- Let K be a compact set in 𝑌 which is a 𝑇 -space, so it is closed set (by remark (1.6)), so 𝑓 K is 𝜔-closed (respectively 𝑁-closed) set (since 𝑓 is an 𝑁-continuous function), so 𝑓 is an M𝜔c-function and an M𝑁c-function. By the same way we can proof the rest. Definition (1.29) A function 𝑓 ∶ 𝑋 ⟶ 𝑌 is called:- 1- A compact function if the inverse image of any compact set in 𝑌 is a compact set in 𝑋 [2]. 2- An 𝜔∗-compact (respectively 𝑁 ∗-compact) function if the inverse image of any compact set in 𝑌 is an 𝜔-compact (respectively 𝑁-compact) set in 𝑋. 3- An 𝜔-compact [12]. (respectively 𝑁-compact) function the inverse image of any 𝜔- compact (respectively 𝑁-compact) set in 𝑌 is a compact set in 𝑋. 4- An 𝜔∗∗-compact (respectively 𝑁∗∗-compact) function the inverse image of any 𝜔-compact (respectively 𝑁-compact) set in 𝑌 is an 𝜔-compact (respectively 𝑁-compact) set in 𝑋. (1.30) Example 1- 𝑓: ℛ, Ʈ ⟶ ℛ, Ʈ is compact, 𝜔∗-compact, 𝑁∗-compact, 𝜔-compact, 𝑁-compact, 𝜔∗∗-compact and 𝑁∗∗-compact function. Proposition (1.31): If 𝑓: 𝑋 ⟶ 𝑌 is:- 1- 𝜔-compact (respectively 𝑁-compact) function where 𝑋 is a 𝑇 -space, and then it is 𝜔Mc- function and 𝜔M𝜔c-function (respectively 𝑁Mc-function and 𝑁M𝑁c-function). 2- 𝜔-compact (respectively 𝑁-compact) function where 𝑋 is a Kc-space, and then it is 𝜔Mc- function and 𝜔M𝜔c-function (respectively 𝑁Mc-function and 𝑁M𝑁c-function). 3- 𝜔-compact (respectively 𝑁-compact) function where 𝑋 is a c-c space, and then it is 𝜔Mc- function and 𝜔M𝜔c-function (respectively 𝑁Mc-function and 𝑁M𝑁c-function). 4- Compact function where 𝑋 is a 𝑇 -space (respectively Kc-space, c-c space), and then it is 𝜔Mc-function and 𝑁Mc-function. 5- 𝜔∗∗-compact (respectively 𝑁∗∗-compact) function where X is a T -space, and then it is ωMωc-function (respectively 𝑁M𝑁c-function). 6- 𝜔∗∗-compact (respectively 𝑁∗∗-compact) function where 𝑋 is a Kc-space, and then it is 𝜔M𝜔c- function (respectively 𝑁M𝑁c-function). 7- 𝜔∗∗-compact (respectively 𝑁∗∗-compact) function where 𝑋 is a c-c space, and then it is 𝜔M𝜔c-function (respectively 𝑁M𝑁c-function). 8- 𝜔∗-compact (respectively 𝑁∗-compact) function where 𝑋 is a 𝜔𝑇 -space (respectively 𝑁𝑇 - space), and then it is M𝜔c-function (resp. M𝑁c-function). 9- 𝜔∗-compact (respectively 𝑁∗-compact) function where 𝑋 is a Kc-space, and then it is M𝜔c-function (respectively M𝑁c-function). 10- 𝜔∗-compact (respectively 𝑁∗-compact) function where 𝑋 is a c-c space, and then it is M𝜔c-function (respectively M𝑁c-function). 11- 𝜔∗∗-compact (respectively 𝑁∗∗-compact) function where 𝑋 is a 𝜔𝑇 -space (respectively 𝑁𝑇 -space), and then it is 𝜔M𝜔c-function (respectively 𝑁M𝑁c-function). Proof: Suppose 𝑓 is 𝜔-compact (respectively 𝑁-compact) function and 𝑋 is a 𝑇 -space, let 𝒲 be an 𝜔-compact set in 𝑌, so 𝑓 𝒲 is compact set in 𝑋 which is a 𝑇 -space, thus 𝑓 𝒲 is   181  Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (3) 2020 closed set (by remark (1.6)), therefore 𝑓 is 𝜔Mc-function (respectively 𝑁Mc-function). Now, since every closed set is 𝑁-closed and 𝜔-closed set, so 𝑓 is 𝜔M𝜔c-function (respectively 𝑁M𝑁c-function). We can prove the rest by the same way. Proposition (1.32) [19]. 1- If 𝑌 is a subspace of 𝑋, Ʈ and 𝒲 is 𝜔-open set in 𝑋, then 𝒲 is 𝜔-open set in 𝑌 provided that 𝒲 ⊆ 𝑌. 2- If 𝑌 is a subspace of 𝑋, Ʈ and 𝒲 is 𝜔-closed set in 𝑋, then 𝒲 is 𝜔-closed set in 𝑌 provided that 𝒲 ⊆ 𝑌. Proposition (1.33) If 𝑓: 𝑋 ⟶ 𝑌 is a 𝜔Mc-function (respectively 𝑁Mc-function) and 𝒲 ⊆ 𝑋 then 𝑓|𝒲 : 𝒲 ⟶ 𝑌 is a 𝜔Mc-function (respectively 𝑁Mc-function). Proof: Let 𝒢=𝑓|𝒲 : 𝒲 ⟶ 𝑌 and let K be an 𝜔-compact (respectively 𝑁-compact) subset of 𝑌, since 𝑓 is an 𝜔Mc-function (respectively 𝑁Mc-function), then 𝑓 𝐾 is a closed subset of 𝑋. Now 𝒢 (K) = 𝑓 (K)⋂𝒲, hence 𝒢 (K) is a closed subset of 𝒲. Therefore 𝒢 𝑓|𝒲 : 𝒲 ⟶ 𝑌 is a 𝜔Mc-function (respectively 𝑁Mc-function). By the same way we can prove the following proposition. Proposition (1.34) If 𝒲 ⊆ 𝑋 and 𝑓: 𝑋 ⟶ 𝑌 is:- 1- An M𝜔c-function (respectively M𝑁c-function), then 𝑓|𝒲 : 𝒲 ⟶ 𝑌 is an M𝜔c-function (respectively M𝑁c-function). 2- An 𝜔M𝜔c-function (respectively 𝑁M𝑁c-function), then 𝑓|𝒲 : 𝒲 ⟶ 𝑌 is a 𝜔M𝜔c- function (respectively 𝑁M𝑁c-function). Proposition (1.35) If 𝑓: 𝑋 ⟶ 𝑌 is 𝜔Mc-function (respectively 𝑁Mc-function) where 𝑋 is a compact space, then it is 𝜔-compact (respectively 𝑁-compact) function. Proof: Suppose K is a 𝜔-compact (respectively 𝑁-compact) subset of 𝑌, so 𝑓 (K) is a closed subset of 𝑋(since 𝑓 is 𝜔Mc-function (respectively 𝑁Mc-function)), but 𝑋 is compact, hence 𝑓 (K) is compact (by remark (1.6)), therefore 𝑓 is 𝜔-compact (respectively 𝑁- compact) function. In a same manner, we can prove the following corollary. Corollary (1.36) If 𝑓: 𝑋 ⟶ 𝑌 is:- 1- 𝜔Mc-function (respectively 𝑁Mc-function) where 𝑋 is a c-c space, then it is 𝜔-compact (respectively 𝑁-compact) function. 2- Mc-function where 𝑋 is a compact space, then it is 𝜔-compact (respectively 𝑁-compact) function. 3- Mc-function where 𝑋 is a c-c space, then it is 𝜔-compact (respectively 𝑁-compact) function. 4- Mc-function where 𝑋 is an 𝜔-compact (respectively 𝑁-compact) space, then it is 𝜔∗- compact (respectively 𝑁 ∗-compact) function. 5- M𝜔c-function (respectively M𝑁c-function) where 𝑋 is an 𝜔-compact (respectively 𝑁- compact) space, then it is 𝜔∗-compact (respectively 𝑁∗-compact) function.   182  Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (3) 2020 6- 𝜔M𝜔c-function (respectively 𝑁M𝑁c-function) where 𝑋 is an 𝜔-compact (respectively 𝑁- compact) space, then it is 𝜔∗∗-compact (respectively 𝑁∗∗-compact) function. 7- M𝜔c-function (respectively M𝑁c-function) where 𝑋 is an 𝜔-compact (respectively 𝑁- compact) space, then it is 𝜔∗∗-compact (respectively 𝑁∗∗-compact) function. 8- Mc-function where 𝑋 is an 𝜔-compact (respectively 𝑁-compact) space, then it is 𝜔∗∗- compact (respectively 𝑁 ∗∗-compact) function. Proof 1- Suppose K is a 𝜔-compact (respectively 𝑁-compact) subset of 𝑌, so 𝑓 (K) is a closed subset of 𝑋(since 𝑓 is 𝜔Mc-function (respectively 𝑁Mc-function)), but 𝑋 is c-c space, hence 𝑓 (K) is compact, therefore 𝑓 is 𝜔-compact (respectively 𝑁-compact) function. 2- Suppose K is a 𝜔-compact (respectively 𝑁-compact) subset of 𝑌, so it is compact (by remark (1.2)), hence 𝑓 (K) is a closed subset of 𝑋(since 𝑓 is Mc-function), but 𝑋 is compact space, hence 𝑓 (K) is compact (by remark (1.6)), therefore 𝑓 is 𝜔-compact (respectively 𝑁- compact) function. 3- Suppose K is an 𝜔-compact (respectively 𝑁-compact) subset of 𝑌, so it is compact (by remark (1.2)), hence 𝑓 (K) is a closed subset of 𝑋(since 𝑓 is Mc-function), but 𝑋 is c-c space, hence 𝑓 (K) is compact, therefore 𝑓 is 𝜔-compact (respectively 𝑁-compact) function. 4- Suppose K is a compact subset of 𝑌, hence 𝑓 (K) is a closed subset of 𝑋(since 𝑓 is Mc- function), and by remark (1.2) it is 𝜔-closed (respectively 𝑁-closed) subset of 𝑋 which is 𝜔- compact (respectively 𝑁-compact) space, hence 𝑓 (K) is 𝜔-compact (respectively 𝑁- compact) (by remark (1.6)), therefore 𝑓 is 𝜔∗-compact (respectively 𝑁∗-compact) function. 5- Suppose K is a compact subset of 𝑌, hence 𝑓 (K) is an 𝜔-closed (respectively 𝑁-closed) subset of 𝑋 ( since 𝑓 is M𝜔c-function (respectively M𝑁c-function)), but 𝑋 is 𝜔-compact (respectively 𝑁-compact) space, hence 𝑓 (K) is 𝜔-compact (respectively 𝑁-compact) (by remark (1.6)), therefore 𝑓 is 𝜔∗-compact (respectively 𝑁 ∗-compact) function. 6- Suppose K is an 𝜔-compact (respectively 𝑁-compact) subset of 𝑌, hence 𝑓 (K) is an 𝜔- closed (respectively 𝑁-closed) subset of 𝑋 ( since 𝑓 is 𝜔M𝜔c-function (respectively 𝑁M𝑁c- function)), but 𝑋 is an 𝜔-compact (respectively 𝑁-compact) space, hence 𝑓 (K) is 𝜔- compact (respectively 𝑁-compact) (by remark (1.6)), therefore 𝑓 is 𝜔∗∗-compact (respectively 𝑁∗∗-compact) function. 7- Suppose K is an 𝜔-compact (respectively 𝑁-compact) subset of 𝑌, so it is compact (by remark (1.2)), hence 𝑓 (K) is an 𝜔-closed (respectively 𝑁-closed) subset of 𝑋 ( since 𝑓 is M𝜔c-function (respectively M𝑁c-function)), but 𝑋 is 𝜔-compact (respectively 𝑁-compact) space, hence 𝑓 (K) is 𝜔-compact (respectively 𝑁-compact) (by remark (1.6)), therefore 𝑓 is 𝜔∗∗-compact (respectively 𝑁∗∗-compact) function. 8- Suppose K is an 𝜔-compact (respectively 𝑁-compact) subset of 𝑌, so it is compact (by remark (1.2)), hence 𝑓 (K) is closed subset of 𝑋 (since 𝑓 is Mc-function (respectively Mc- function)), and by remark (1.2)) it is an 𝜔-closed (respectively 𝑁-closed), but 𝑋 is 𝜔-compact (respectively 𝑁-compact) space, hence 𝑓 (K) is 𝜔-compact (respectively 𝑁-compact) (by remark (1.6)), therefore 𝑓 is 𝜔∗∗-compact (respectively 𝑁∗∗-compact) function. Proposition (1.37) If 𝑋 is an 𝜔-compact and 𝜔𝑇 -space (respectively 𝑁-compact and 𝑁𝑇 -space), then a function 𝑓: 𝑋 ⟶ 𝑌 is an 𝜔∗∗-compact (respectively 𝑁 ∗∗-compact) function iff it is 𝜔M𝜔c- function (respectively 𝑁M𝑁c-function).   183  Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (3) 2020 Proof: Let 𝑋 be an 𝜔-compact and 𝜔𝑇 -space (respectively 𝑁-compact and 𝑁𝑇 -space), let 𝑓 be an 𝜔∗∗-compact (respectively 𝑁∗∗-compact). To prove 𝑓 is an 𝜔M𝜔c-function (respectively 𝑁M𝑁c-function). Let ℳ be an 𝜔-compact (respectively 𝑁-compact) set in 𝑌, then 𝑓 ℳ is 𝜔-compact (respectively 𝑁-compact) set in 𝑋 (since 𝑓 is 𝜔∗∗-compact (respectively 𝑁∗∗- compact)), and since 𝑋 is 𝜔𝑇 -space (respectively 𝑁𝑇 -space), so 𝑓 ℳ is 𝜔-closed (respectively 𝑁-closed) set in 𝑋, hence 𝑓 is an 𝜔M𝜔c-funtion (respectively 𝑁M𝑁c-function). Conversely, let 𝑓 be an 𝜔M𝜔c-function (respectively 𝑁M𝑁c-function). To prove 𝑓 is an 𝜔∗∗- compact (respectively 𝑁 ∗∗-compact). Let ℳ be an 𝜔-compact (respectively 𝑁-compact) set in 𝑌, so 𝑓 ℳ) is 𝜔-closed (respectively 𝑁-closed) set in 𝑋 (because 𝑓 is 𝜔M𝜔c-function (respectively 𝑁M𝑁c-function)), but 𝑋 is 𝜔-compact (respectively. 𝑁-compact) space, hence 𝑓 ℳ is 𝜔-compact (respectively 𝑁-compact) set, therefore 𝑓 is 𝜔∗∗-compact (respectively 𝑁∗∗-compact) function. Corollary (1.38)   1‐ If 𝑋 is a Kc-space, and 𝑓: 𝑋 ⟶ 𝑌 is an 𝜔∗∗-compact (respectively 𝑁∗∗-compact) function, then it is an 𝜔M𝜔c-function (respectively 𝑁M𝑁c-function). 2- If 𝑋 is an 𝜔-compact (respectively 𝑁-compact) space and Kc-space, then a function 𝑓: 𝑋 ⟶ 𝑌 is an 𝜔∗∗-compact (respectively 𝑁∗∗-compact) function iff it is 𝜔M𝜔c-function (respectively 𝑁M𝑁c-function). 3- If 𝑋 is a c-c space, and 𝑓: 𝑋 ⟶ 𝑌 is an 𝜔∗∗-compact (respectively 𝑁∗∗-compact) function, then it is an 𝜔M𝜔c-function (respectively 𝑁M𝑁c-function). 4- If 𝑋 is an 𝜔-compact (respectively 𝑁-compact) and c-c space, then a function 𝑓: 𝑋 ⟶ 𝑌 is an 𝜔∗∗-compact (respectively 𝑁 ∗∗-compact) function iff it is 𝜔M𝜔c-function (respectively 𝑁M𝑁c-function). 5- If 𝑋 is an 𝜔-compact (respectively 𝑁-compact) and 𝑇 -space, then a function 𝑓: 𝑋 ⟶ 𝑌 is an 𝜔∗∗-compact (respectively 𝑁 ∗∗-compact) function iff it is 𝜔M𝜔c-function (respectively 𝑁M𝑁c-function). Proof: 1- Let K be an 𝜔-compact (respectively 𝑁-compact) subset of 𝑌, so 𝑓 (K) is 𝜔-compact (respectively 𝑁-compact) subset of 𝑋 (since 𝑓 is an 𝜔∗∗-compact (respectively 𝑁 ∗∗-compact) function)), and then it is compact subset of 𝑋 (by remark (1.2)) which is Kc-space, so 𝑓 (K) is closed subset of 𝑋, and by remark (1.2)) it is 𝜔-closed (respectively 𝑁-closed), so 𝑓 is an 𝜔M𝜔c-function (respectively 𝑁M𝑁c-function). 2- Let 𝑓 be an 𝜔∗∗-compact (respectively 𝑁∗∗-compact) function, and K be an 𝜔-compact (respectively 𝑁-compact) subset of 𝑌, so 𝑓 (K) is 𝜔-compact (respectively 𝑁-compact) subset of 𝑋 (since 𝑓 is an 𝜔∗∗-compact (respectively 𝑁∗∗-compact) function)), and then it is compact subset of 𝑋 (by remark (1.2)) which is Kc-space, so 𝑓 (K) is closed subset of 𝑋, and by remark (1.2)) it is 𝜔-closed (respectively 𝑁-closed), so 𝑓 is an 𝜔M𝜔c-function (respectively 𝑁M𝑁c-function). Conversely, let K be an 𝜔-compact (respectively 𝑁-compact) subset of 𝑌 so 𝑓 (K) is 𝜔-closed (respectively 𝑁-closed) subset of 𝑋 (since 𝑓 is an 𝜔M𝜔c- function (respectively 𝑁M𝑁c-function)), but 𝑋 is an 𝜔-compact (respectively 𝑁-compact) space, so 𝑓 (K) is an 𝜔-compact (respectively 𝑁-compact) subset of 𝑋, therefore 𝑓 is an 𝜔∗∗-compact (respectively 𝑁∗∗-compact) function.   184  Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (3) 2020 3- Let K be an 𝜔-compact (respectively 𝑁-compact) subset of 𝑌, so 𝑓 (K) is 𝜔-compact (respectively 𝑁-compact) subset of 𝑋 (since 𝑓 is an 𝜔∗∗-compact (resp. 𝑁 ∗∗-compact) function)), and then it is compact subset of 𝑋 (by remark (1.2)) which is c-c space, so 𝑓 (K) is closed subset of 𝑋, and by remark (1.2)) it is 𝜔-closed (respectively 𝑁-closed) subset of 𝑋, so 𝑓 is an 𝜔M𝜔c-function (respectively 𝑁M𝑁c-function). 4- Let 𝑓 be an 𝜔∗∗-compact (respectively 𝑁∗∗-compact) function, and K be an 𝜔-compact (respectively 𝑁-compact) subset of 𝑌, so 𝑓 (K) is 𝜔-compact (respectively 𝑁-compact) subset of 𝑋 (since 𝑓 is an 𝜔∗∗-compact (resp. 𝑁∗∗-compact) function)), and then it is compact subset of 𝑋 (by remark (1.2)) which is c-c space, so 𝑓 (K) is closed subset of 𝑋, and by remark (1.2)) it is 𝜔-closed (respectively 𝑁-closed) subset of 𝑋, so 𝑓 is an 𝜔M𝜔c-function (respectively 𝑁M𝑁c-function). Conversely, let K be an 𝜔-compact (respectively 𝑁-compact) subset of 𝑌, so 𝑓 (K) is 𝜔-closed (respectively 𝑁-closed) subset of 𝑋 (since 𝑓 is an 𝜔M𝜔c- function (respectively 𝑁M𝑁c-function) which is an 𝜔-compact (respectively 𝑁-compact) space, so 𝑓 (K) is an 𝜔-compact (respectively 𝑁-compact) subset of 𝑋(by remark (1.6)) , so 𝑓 be an 𝜔∗∗-compact (respectively 𝑁∗∗-compact) function. 5- Let 𝑓 be a 𝜔∗∗-compact (respectively 𝑁∗∗-compact) function, and K be an 𝜔-compact (respectively 𝑁-compact) subset of 𝑌, so 𝑓 (K) is 𝜔-compact (respectively 𝑁-compact) subset of 𝑋 (since 𝑓 is an 𝜔∗∗-compact (resp. 𝑁∗∗-compact) function)), and then it is compact subset of 𝑋 (by remark (1.2)) which is a 𝑇 -space, so 𝑓 (K) is closed subset of 𝑋 (by remark (1.6)), and by remark (1.2)) it is 𝜔-closed (respectively 𝑁-closed) subset of 𝑋, so 𝑓 is an 𝜔M𝜔c-function (respectively 𝑁M𝑁c-function). Conversely, let K be a 𝜔-compact (respectively 𝑁-compact) subset of 𝑌, so 𝑓 (K) is 𝜔-closed (respectively 𝑁-closed) subset of 𝑋 (since 𝑓 is an 𝜔M𝜔c-function (respectively 𝑁M𝑁c-function) which is an 𝜔-compact (respectively 𝑁-compact) space, so 𝑓 (K) is a 𝜔-compact (respectively 𝑁-compact) subset of 𝑋(by remark (1.6)) , so 𝑓 be an 𝜔∗∗-compact (respectively 𝑁∗∗-compact) function. Proposition (1.39) The composition between: 1- Strongly 𝜔-continuous function and M𝜔c-function (respectively M𝑁c-function) is Mc- function. 2- 𝜔-continuous (respectively 𝑁-continuous) function and 𝜔Mc-function (respectively 𝑁Mc- function) is 𝜔M𝜔c-function (respectively 𝑁M𝑁c-function). 3- Strongly 𝜔-continuous (respectively strongly 𝑁-continuous) function and 𝜔M𝜔c-function (respectively 𝑁M𝑁c-function) is 𝜔Mc-function (respectively 𝑁Mc-function). 4- Strongly 𝜔-continuous (respectively strongly 𝑁-continuous) function and Mc-function is Mc-function. 5- Strongly 𝜔-continuous (respectively strongly 𝑁-continuous) function and 𝜔Mc-function (respectively 𝑁Mc-function) is 𝜔Mc-function (respectively 𝑁Mc-function). 6- Strongly 𝜔-continuous (respectively strongly 𝑁-continuous) function and M𝜔c-function (respectively M𝑁c-function) is Mc-function. Proof: 1- Let 𝑓: 𝑋 ⟶ 𝑌 be strongly 𝜔-continuous function and 𝒢: 𝑌 ⟶ 𝒵 be M𝜔c-function (resp. M𝑁c-function), and let K be a compact set in 𝒵, so 𝒢 K is 𝜔-closed (respectively 𝑁- closed) set in 𝑌, so 𝑓 (𝒢 K = 𝒢 ∘ 𝑓 (K) is closed set in 𝑋(since every 𝑁-closed set is 𝜔-closed), so 𝒢 ∘ 𝑓 is Mc-function.   185  Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (3) 2020 2- Let 𝑓: 𝑋 ⟶ 𝑌 be an 𝜔-continuous (respectively 𝑁-continuous) function and 𝒢: 𝑌 ⟶ 𝒵 be 𝜔Mc-function (respectively 𝑁Mc-function), and let K be an 𝜔-compact (respectively 𝑁- compact) set in 𝒵, so 𝒢 K is closed set in 𝑌, and then 𝑓 (𝒢 K = 𝒢 ∘ 𝑓 (K) is 𝜔- closed (respectively 𝑁-closed) set in 𝑋, so 𝒢 ∘ 𝑓 is 𝜔M𝜔c-function (resp. 𝑁M𝑁c-function). 3- Let 𝑓: 𝑋 ⟶ 𝑌 be strongly 𝜔-continuous (respectively strongly 𝑁-continuous) function and 𝒢: 𝑌 ⟶ 𝒵 be an 𝜔M𝜔c-function (respectively 𝑁M𝑁c-function), and let K be an 𝜔-compact (respectively 𝑁-compact) set in 𝒵, so 𝒢 K is 𝜔-closed (respectively 𝑁-closed) set in 𝑌, so 𝑓 (𝒢 K = 𝒢 ∘ 𝑓 (K) is closed set in 𝑋(since every 𝑁-closed set is 𝜔-closed), so 𝒢 ∘ 𝑓 is 𝜔Mc-function (respectively 𝑁Mc-function). 4- Let 𝑓: 𝑋 ⟶ 𝑌 be strongly 𝜔-continuous (respectively strongly 𝑁-continuous) function and 𝒢: 𝑌 ⟶ 𝒵 be Mc-function, and let K be a compact set in 𝒵, so 𝒢 K is closed set in 𝑌, and by remark (1.2) it is 𝜔-closed (respectively 𝑁-closed) set in 𝑌, so 𝑓 (𝒢 K = 𝒢 ∘ 𝑓 (K) is closed set in 𝑋, so 𝒢 ∘ 𝑓 is Mc-function. 5- Let 𝑓: 𝑋 ⟶ 𝑌 be strongly 𝜔-continuous (respectively strongly 𝑁-continuous) function and 𝒢: 𝑌 ⟶ 𝒵 be 𝜔Mc-function (respectively 𝑁Mc-function), and let K be an 𝜔-compact (respectively 𝑁-compact) set in 𝒵, so 𝒢 K is closed set in 𝑌, and by remark (1.2) it is 𝜔- closed (respectively 𝑁-closed) set in 𝑌, so 𝑓 (𝒢 K = 𝒢 ∘ 𝑓 (K) is closed set in 𝑋, so 𝒢 ∘ 𝑓 is 𝜔Mc-function (resp. 𝑁Mc-function). 6- Let 𝑓: 𝑋 ⟶ 𝑌 be strongly 𝜔-continuous (respectively strongly 𝑁-continuous) function and 𝒢: 𝑌 ⟶ 𝒵 be M𝜔c-function (respectively M𝑁c-function), and let K be a compact set in 𝒵, so 𝒢 K is 𝜔-closed (respectively 𝑁-closed) set in 𝑌, so 𝑓 (𝒢 K = 𝒢 ∘ 𝑓 (K) is closed set in 𝑋 (since every 𝑁-closed set is 𝜔-closed), so 𝒢 ∘ 𝑓 is Mc-function. Proposition (1.40) If 𝑓: 𝑋 ⟶ 𝑌 and 𝒢: 𝑌 ⟶ 𝒵 are two functions:- 1- If 𝑓 is M𝜔c-function (respectively M𝑁c-function) and 𝒢 is strongly 𝜔-continuous function where 𝑌 is a compact space, then 𝒢 ∘ 𝑓 is an 𝜔-irresolute (respectively 𝑁-irresolute) function. 2- If 𝑓 is 𝜔Mc-function (respectively 𝑁Mc-function) and 𝒢 is 𝜔-continuous (respectively 𝑁- continuous) function where 𝑌 is a 𝜔-compact (respectively 𝑁-compact) space, then 𝒢 ∘ 𝑓 is a continuous function. 3- If 𝑓 is 𝜔M𝜔c-function (respectively 𝑁M𝑁c-function) and 𝒢 is strongly 𝜔-continuous (respectively strongly 𝑁-continuous) function where 𝑌 is an 𝜔-compact (respectively 𝑁- compact) space, then 𝒢 ∘ 𝑓 is an 𝜔-irresolute (respectively 𝑁-irresolute) function. 4- If 𝑓 is Mc-function and 𝒢 is 𝜔Mc-function (respectively 𝑁Mc-function) function where 𝑌 is a compact space, then 𝒢 ∘ 𝑓 is 𝜔Mc-function (respectively 𝑁Mc-function). 5- If 𝑓 is a compact function and 𝒢 is 𝜔Mc-function (respectively 𝑁Mc-function) where 𝑌 is a compact space, then 𝒢 ∘ 𝑓 is a 𝜔-compact (respectively 𝑁-compact) function 6- If 𝑓 is Mc-function and 𝒢 is strongly 𝜔-continuous (respectively strongly 𝑁-continuous) function where 𝑌 is a c-c space, then 𝒢 ∘ 𝑓 is strongly 𝜔-continuous (respectively strongly 𝑁- continuous) function. 7- If 𝑓 is M𝜔c-function (respectively M𝑁c-function) and 𝒢 is strongly 𝜔-continuous function where 𝑌 is a c-c space, then 𝒢 ∘ 𝑓 is an 𝜔-irresolute (respectively 𝑁-irresolute) function. And there are many other cases.   186  Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (3) 2020 Proof: 1- Let 𝑓 be an M𝜔c-function (respectively M𝑁c-function) and 𝒢 be a strongly 𝜔-continuous function, and let 𝒲 be 𝜔-open set in 𝒵, so 𝒢 𝒲 is open set in 𝑌, so (𝒢 𝒲 is closed set in 𝑌 which is compact space, so 𝒢 𝒲 is compact set in 𝑌, then 𝑓 𝒢 𝒲) ]=[ 𝒢 ∘ 𝑓 𝒲 is 𝜔-closed (respectively 𝑁-closed) set in 𝑋 (since 𝑓 𝒢 𝒲 ] =𝑓 𝑌 𝒢 𝒲 𝑋 𝑓 𝒢 𝒲 𝑋 𝒢 ∘ 𝑓 𝒲 𝒢 ∘ 𝑓 𝒲 ), and then 𝒢 ∘ 𝑓 𝒲 is 𝜔-open (respectively 𝑁-open) set in 𝑋, hence 𝒢 ∘ 𝑓 is 𝜔-irresolute (respectively 𝑁-irresolute) function. By the same way we can prove the rest. References 1. Farhan, A.A.M. On-MC-Functions, Tikrit Journal of Pure Science.2011, 16, 4, 287-289. 2. AL-Saidy, S.K.; Abtan, R.G.; Ali, H.J. More on MC-Functions, Journal of the College of Basic Education.2015, 21, 89, 149-156. 3. Hdeib, H.Z. 𝜔-closed mappings, Revista Colombiana de Matemáticas.1982, 16, 1-2, 65- 78. 4. Al-Omari, A.H.M.A.D.; Noorani, M.M. 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