Nb-Compact and Nb-Lindelöf Spaces

In this work, we present new types of compact and Lindelöf spaces and some facts and results related to them. There are also types of compact and Lindelöf functions and the relationship between them has been investigated. Further, we have present some properties and results related to them


2-Preliminaries
Definition2.1 [2]: If is part of a space X that is -open, then .
Definition2.2 [3]: A subset A of a space X is said to be an N-open if for every p∈A there exists an open subset Up in X such that Up-A is a finite set.
The complement of an N-open set is said to be N-closed.Definition2.4[2]: A topological space X is said to be b-compact if every b-open cover of X has a finite subcover.
Definition2.6:[11] X is nearly compact if every regular open cover for X reduced to a finite subcover.
Definition2.7:[11] If every regular open set cover of X has a countably sub-cover, it becomes nearly Lindelöf.

Definition2.8 [3]:
A topological space X is said to be N-compact if every N-open cover of X has a finite subcover.
Definition2.9:[12] A topological space X is said to be b-Lindelöf if every b-open cover of X has a countable subcover.

Definition2
. 10 [11]: A topological space X is said to be nearly Lindelöf if every regular open cover of X has a countable subcover.
Definition2.11:[12] A topological space X is said to be nearly b-Lindelöf if every b-regular open cover of X has a countable subcover.

Definition2.12:
A topological space X is said to be nearly N-Lindelöf if every N-regular open cover of X has a countable subcover.

3-Nb-compact spaces
We will explore a novel type of open sets of Nb-compact, we need the followed definitions Definition3.

Lemma 3.10:
Lemma 3.12: [6] Let(X, ) serve as a topological space;   Definition 3.20: [7] Let f:X→Y be function of a topological space (X,τ) into a topological space( ) then f is referred to be a Nb-irresolute function if

Definition 3.24:
A collection C of sets is said to have finite-intersection-property iff the intersection of members of each finite sub-collection of C is non-empty.Proposition 3.25: A topological space (X, ) is Nb-compact iff any collection of Nb-closed subsets of X with finite-intersection-property has a non-empty intersection.10-If f is Lindelöf and g is Nb-Lindelöf, then is Lindelöf.11-If f is Lindelöf and g is Lindelöf gof is Lindelöf Proof: (1) Let L be Lindelöf subset of M, so ( is Nb-Lindelöf in Y (since g is Nb-Lindelöf).Also ( ( ) is Nb-Lindelöf in X (since f is ) but ( ( ) ( ( , then is Nb-Lindelöf.The other by the same way of (1).

Conclusion
In our work, we deduced a strong types of compact and Lindelöf spaces.Also, we obtained some types of weak and strong functions of compact and Lindelöf functions by using open sets of type Nb which will be powerful formulas to concepts Nb-compact and Nb-Lindelöf if defined, which has a direct relationship with the functions and Lindelöf as a future work.
present the concept of Nb-compact and Nb-Lindelöf with definitions, properties and examples based on the two concepts b-open and N-open sets.
A subset A of a space X is said to be an Nb-open set if for each x∈A there exists a b-open set U in X with x∈U and U-A= finite.The complement of Nb-open sets is called Nb-closed.

Definition2. 13 :
Let X be topological space and A X, A is called Nb-regular-open set in X if = .the complement of Nb-regular-open set is called -regular-closed thus it is simple to observe that A is Nb-regular closed set if A = .

Remark 3 . 8 :Lemma 3 . 9 :
The relation between N-compact and b-compact is missed.A subset U is Nb-open in X every point in U is an Nb-interior point to U. Proof: Since ∈ so x is an Nb-interior point of U and this way identically for all points of U. Conversely, since every Nb-interior point to U then there is Nb-open set contain this point and so ⋃ ∈ , but the random union of Nb-open sets is Nbopen, so U is Nb-open.

Proposition 3 . 13 :
1.A b-open set is created when an open set and a b-open set intersect.2. b-open sets are created by joining any family of them.It is Nb-open when a Nb-open set intersects with a N open set.Proof: A should be an Nb-open set, and B must be an N-open set in space X.Allowing x as any point of .Because of A is Nb-open, a b-open set is available U A comprising x in a way that |U A − A| is finite.Since B is N-open, An unclosed set exists U B comprising x in a way that |U B − B| is finite.By Lemma(3.12),U A U B is b-open set that includes x and (U A U

Corollary 3 . 14 :
An Nb-open set is created when such an open set and another Nb-open set intersect.Proof: Due to the fact that every open set is N-open, the intersection is maintained by the aforementioned proposition.Theorem 3.15: In a N-Hausdorff space, every Nb-compact subset is also a Nb-closed.Proof: Set X be an N-Hausdorff space and be its Nb-compact subset, to demonstrate let , we demonstrate the existence of a N-open set that includes x and is disjoint from Y, in each ∈ it is distinct from x, choose disjoint N-open sets U x and V y contains x and y (respectively) since X is N-Hausdorff, the collection{V y :y∈ is N-open cover which is Nb-open cover to Y but Y is Nb-compact, Consequently, they are limited in number V y1 ,V y2 , . ., V yn the Y cover the N-open set =⋃ yi includes Y and is not coupled to the N-open set U =⋂ xi , by obtaining the intersect of N-open sets that contain x, since if z is a point of V so z∈ yi for a few , hence z xi and z U is N-open so it is an Nbopen set contains x disjoint from Y, then x

Proposition 3 . 28 :
For any topological space (X, ), these two claims are interchangeable: 1 -X is nearly Nb-compact.2-Every Nb regular open cover p = { : ∈ of X, a limited subset exists with p= { : ∈ be Nb-regular open cover for X then { ∈ is Nb-regular open cover for the nearly Nb-compact space X thus a limited subset exists with X =⋃ ∈ ∈ (2) (l) It is clear since Nb-regular open set is Nb-open.Theorem 3.29: For all topological spaces (X, ), these three claims are interchangeable: 1 -X is nearly Nb-compact.2-Any family of Nb-closed sets { : ∈ of X with ⋂ ∈ then a finite subset exists hence ⋂ ∈ 3-Any family of Nb-regular closed sets { : ∈ of X such that ⋂ ∈ then a limited subset exists hence ⋂ ∈ Proof: Let { : ∈ be family of Nb-closed sets of X, with ⋂ ∈ let , the family { : ∈ is an Nb-open cover of space X, since X is nearly Nb-compact by proposition(3.25)there exists a finite subset such that

1 :
The topological space X we name it b-space if every b-open set is open in it.
-1 ( of each Nb-open set A in Y corresponds to a Nb-open set in X. If (X, ) is a topological space and Nb-open subset of X is Nb-compact relative to X then any subset is Nb-compact relative to X.
Definition 3.27: A topological space (X, ) is defined as nearly Nb-compact if each open Nbregular cover of X has a finite subcover.