On regular generalized N−Preopen sets

The purpose of this paper is to provide a new class of generalized N−preopen sets, namely, regular generalized N−preopen sets which is finer than the class of regular generalized preopen sets and the class of regular generalized open sets. Furthermore, we study the fundamental topological properties and introduce the notion of regular generalized N−precontinuous functions.

study its topological properties. Furthermore, the relationship with the other known sets will be studied. In Section 4 we introduce the notion of generalized N−precontinuous functions and approximately N−precontinuous.

Preliminaries
In this section, we provide some preliminary works that serve as background for the present study.
Theorem 2.1. [12] A subset A of a topological space (X, ) is a r−closed set if and only if , A = Cl(Int(A)). Theorem 2.4. [8] Let A and B be two subsets in a topological space (X, ). If A is a preopen set in X and B is an open set in X, then A  B is a preopen set in X.
Theorem 2.5. [8] Let (X, ) and (Y,  ) be two topological spaces. If A×B is a preopen set in (X × Y,  ×  ) if, and only if , A is a preopen set in (X, ) and B is a preopen set in (Y,  ). Lemma 2.6. [3] For a topological space (X, ) and A  X, the following hold: 1. Intn(X − A) = X − Cln(A).

Regular generalized N−preopen sets
Definition 3.1. A subset A of a topological space (X, ) is called regular generalizedN−preclosed set (simply RNg−preclosed), if Cln(A)  U, whenever A  U and U is r−open subset of (X, ).
The complement of RNg−preclosed set is called regular generalized N−preopen set (simply RNg−preopen). It is clear that every Ng−preclosed set is RNg−preclosed set. The converse of this fact need not be true.  Proof. Let A be rg−preclosed subset of a topological space (X, ) and U be any r−open set such that AU. Since A is a rg−preclosed set, Clp(A)  U. Since Cln(A)Clp(A), then Cln(A)U. Therefore, A is a RNg−preclosed set.
In Example (3.3), E2 is RNg−preclosed set which is not Ng−preclosed set. That is, the set 2 = {1} is RNg−preopen set which is not Ng−preopen set. We have the following implications for RNg−preopen set with theses sets:  Proof. Let A be a RNg−preopen subset of X and F be a r−closed subset of X such that F  A. Then Conversely, suppose that F  Intn(A) where F is a r−closed subset of X such that F  A. Then for any r−open subset U of X such that X −A  U, we have X −U  A and X − U  Intn(A). Then by, Lemma(2.6), X − Intn(A) = Cln(X − A)  U. Hence X − A is a RNg−preclosed set.
That is, A is a RNg−preopen set. Proof. Suppose that F be a r−closed subset of X such that F  Cln(A) − A. Then F  X−A and hence A  X − F. Since A is a RNg−preclosed set and X −F is a r−open subset of X, then Cln(A)  X − F and so F  X − Cln(A). Therefore,  That is, B is a RNg−preclosed set.   Proof. Let xA be arbitrary point in A. Since A × B is a nonempty, take yB. Since (x, y)A × B and A × B is a N−preopen set in (X × Y,  ×  ), then there is a preopen set U × G in is a finite, that is, U −A is also a finite. Since U ×G is a preopen in X × Y , then by, Theorem(2.5), U is a preopen set in (X, ) and G is a preopen set in (Y,  ). Hence A is a N−preopen set in (X, ) and similarly, B is a N−preopen set in (Y,  ). Proof. Let (x, y) Intn(A×B). Then , by the definition of Intn(A×B) there is at least one a N−preopen set U × G in X ×Y such that (x, y)  U ×G  A×B. This implies x  U  A and y Univ. Aden J. Nat. and Appl. Sc. Vol. 24 No.1 -April  G  B. Since U × G is a N−preopen set in X × Y , then by Lemma, (3.13), U is a N−preopen set in X and G is a N−preopen set in Y . Then x  Intn(A) and y  Intn(B). This implies (x, y)  Intn(A) × Intn(B). Therefore
Proof. Let F1 be a r−closed set in X and F2 be a r−closed set in Y such that F1  A and F2  B. By Theorem(2.3), F1 × F2 is a r−closed set in X × Y . Since A × B is a nonempty RNg−preopen set in (X × Y,  ×  ) and by Lemma (3.14), then This implies F1  Intn(A) and F2  Intn(B). Hence A is a RNg−preopen set in (X,  ) and B is a RNg−preopen set in (Y,  ).    It is clear that every rg−precontinuous function is RNg−precontinuous function and the converse of this fact need not be true.  Proof. Let f : (X,  ) → (Y,  ) be a N−precontinuous and F be any closed set in Y .

Regular generalized N −precontinuous functions:
Conversely, suppose that be any open set in Y . Then, by the hypothesis, Conversely, suppose that every singleton set in X is either r−closed or N−preopen. Let A be any RNg−preclosed set in X and x  Cln(A). We show that x A. By the Hypothesis, {x} is either