On new structure of N-topology

Indeed, the bitopological space propounded and introduced by Kelly in the year 1963, kept haunting our Mathematical mind. He introduced the bitopological space which is a non empty set X equipped with two arbitrary topologies 1 and 2 . In this space, the open sets are called pairwise open sets. In this paper, we establish bitopological space with bitopological axioms and prove the structure of a non *Corresponding author: M. Lellis Thivagar, School of Mathematics, Madurai Kamaraj University, Madurai 625021, Tamil Nadu, India E-mail: kabilanchelian@gmail.com


Introduction
The intrinsic nature and beauty of Mathematics is this: One must be in "love" with Mathematics. As a result, the nature of inquisitiveness in a person gets, needless to mention, always enkindled and triggered by the new theorems or axioms or any new findings, even if it is a small in its nature or incredibly big.
Indeed, the bitopological space propounded and introduced by Kelly in the year 1963, kept haunting our Mathematical mind. He introduced the bitopological space which is a non empty set X equipped with two arbitrary topologies 1 and 2 . In this space, the open sets are called pairwise open sets. In this paper, we establish bitopological space with bitopological axioms and prove the structure of a non ABOUT THE AUTHORS M. Lellis Thivagar has published 210 research publications both in national and International journals to his credit. Under his able guidance 15 scholars obtained their doctoral degree. In his collaborative work, he has joined hands with intellectuals of highly reputed persons internationally. He serves as a referee for 12 peerreviewed international journals. At present he is the professor, chairperson, School of Mathematics, Madurai Kamaraj University.
V. Ramesh is a research scholar under the guidance of M. Lellis Thivagar at the School of Mathematics, Madurai Kamaraj University, Madurai. Five of his research papers are published/ accepted in the reputed international peerreviewed journals.
M. Arockia Dasan is also a research scholar under the guidance of M. Lellis Thivagar at the School of Mathematics, Madurai Kamaraj University, Madurai. Four of his research papers are published/accepted in the reputed international peer-reviewed journals.

PUBLIC INTEREST STATEMENT
The intrinsic nature and beauty of Mathematics is this: it keeps growing within. It manifests its own manifold beauties, in an exponential quotient, when persons evince keen enthusiasm and starts grappling with and further fathom into its colorful nature and its features. So far, as much we are aware of the publications and other writings in vogue, we may be the first ones, who have tried, herewith, to establish bitopological space with bitopological axioms and proved the structure of non empty set X equipped with more than two topologies. Here, we have defined the structure of N-topology, that is, a non empty set X equipped with N-arbitrary topologies, and which has its own open sets. Further, we introduce continuous functions on N-topological space which in turn has its own impact on the Pasting Lemma.
empty set X equipped with more than two topologies. Recently many researchers defined various forms of open sets in this space such as 1 2 (Lellis Thivagar, 1991), 1,2 (Lellis Thivagar, Ekici, & Ravi, 2008), etc. In addition to our fervent efforts, herein, we have also tried to prove the structure of Ntopology, that is, a non empty set X equipped with N-arbitrary topologies 1 , 2 , … , N and also established its own open sets. Further, we study its characterizations. Also, we introduce continuous functions on such topological spaces and establish their basic properties and proved the Pasting Lemma.

Preliminaries
Definition 2.1 (Doitchinov, 1988) A quasi-pseudo-metric on a non empty set X is a function where ℝ + is the set of all positive real numbers.
Definition 2.2 (Grabiec, Cho & Saadati, 2007) Let d 1 a quasi-pseudo-metric on X, and let a function Trivially d 2 is a quasi-pseudometric defined on X and we say that d 1 and d 2 are conjugate one another.
the open d 1 -sphere with centre x and radius k 1 > 0. Classically, the collection of all open d 1 -spheres forms a base for a topology, the obtained topology, be denoted by 1 and called the quasi-pseudo-metric topology of d 1 . Similarly we get a topology 2 for X, due to the quasi-pseudo-metric d 2 .
Definition 2.3 (Kelly, 1963) A non empty set X equipped with two arbitrary topologies 1 and 2 is called a bitopological space and is denoted by (X, 1 , 2 ).

N-topological spaces
In this section, we introduce the notion of N-topological spaces and its own open sets. We derive its basic properties. We also define and discuss the relative topology in N-topological spaces.
Definition 3.1 Let d 1 and d 2 be conjugate, quasi-pseudo-metrics on X and define a function Therefore, d 3 is a quasi-pseudo-metric on X and which is called a Mean Conjugate (simply write M.C) of d 1 , d 2 and d 1 . For each i = 1, 2, 3, the quasi-pseudo metric d i gives a topology i whose base is Thus we define a non empty set X equipped with three arbitrary topologies 1 , 2 and 3 is called a tritopological space and is denoted by (X, 3 ) or (X, 1 , 2 , 3 ).
We can easily verify that d N is a quasi-pseudo-metric on X. Also we note that for each N, d N (x, y) ≠ d N (y, x) for all x, y ∈ X and d N is called a Mean Conjugate (simply write M.C) of d 1 , d 2 , ..., d N−1 and d 1 . For each i = 1, … , N, the quasi-pseudo metric d i gives a topology i whose base is Thus we define a non empty set X equipped with N-arbitrary topologies 1 , 2 , ..., and N is called a N-topological space and is denoted by (X, N ) or (X, 1 , 2 , … , N ).
Definition 3.2 Let X be a non empty set, 1 and 2 be two arbitrary topologies defined on X and the collection 2 be defined by satisfying the following axioms: Then the pair (X, 2 ) is called a bitopological space on X and the elements of the collection 2 are known as 2 -open sets on X.
We can generalize the above definition as given below: let X be a non empty set, 1 , 2 , ..., N be N-arbitrary topologies defined on X and let the collection N be defined by satisfying the following axioms: Then the pair (X, N ) is called a N-topological space on X and the elements of the collection N are known as N -open sets on X. A subset A of X is said to be N -closed on X if the complement of A is Nopen on X. The set of all N -open sets on X and the set of all N -closed sets on X are, respectively, denoted by N O(X) and N C(X). �, {a}, {d}, {a, d}, {b, d}, {c, d}, {a, b, d}, {a, c, d}, {b, c, d}} �, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c}, {b, c, d}} (ii) Intersection of two 2 is also a 2 .
Intersection of two 3 is also a 3 .
In general, intersection of two N-topology is again a N-topology.
for all x, y ∈ X.
Remark 3.5 Union of two 2 need not be a 2 . Union of two 3 need not be a 3 .
In general, union of two N-topology need not be a N-topology.
Definition 3.7 Let X be a non empty set and S be a subset of X. Then (i) (a) The 2 -interior of S, denoted by 2 -int(S), and is defined by The 3 -interior of S, denoted by 3 -int(S), and is defined by (c) Generally, the N -interior of S, denoted by N -int(S), and is defined by (ii) (a) The 2 -closure of S, denoted by 2 -cl(S), and is defined by (b) The 3 -closure of S, denoted by 3 -cl(S), and is defined by (c) Generally, the N -closure of S, denoted by N -cl(S), and is defined by Conversely, assume N -cl(A) = A. Then A is the smallest N -closed set containing itself. Therefore, A is N -closed. Particularly, since ∅ and X are N -closed sets, then N -cl(�) = � and N -cl(X) = X. Theorem 3.10 Let (X, N ) be a N-topological space on X. Then N -closure satisfies Kuratowski closure axioms given below: Proof Proof is follows from (i), (ii), (iv) and (vi) of theorem 3.8. ✷ Proof Assume x ∈ N -cl(A) and G is a N -open set containing x, then X − G is N -closed set and which is a contradiction. Therefore, x ∈ N -cl(A). ✷ Theorem 3.12 Let (X, N ) be a N-topological space X and A ⊆ X.
Theorem 3.20 Let (Y, (N ) * ) be a subspace of (X, N ) and A ⊆ Y. Then

Continuity in N-topological spaces
In this section, we introduce continuous functions in N-topological spaces and discuss the different properties of it. Also, we prove the Pasting Lemma. Throughout this section, the N-topological spaces (X, N ) and (Y, N ) represented by X and Y, respectively.
Therefore, f is N * -continuous function on X. ✷ , even though f is 2 * -continuous function on X. That is, equality does not hold in the theorem 4.4, even though f is 2 * -continuous function on X.
, even though f is 2 * -continuous function on X. That is, equality does not hold in the theorem 4.6, even though f is 2 * -continuous function on X. ).
for every B ⊆ Y and let G be a N -open set in Y. Then N -int(G) = G and by assumption, Proof Let F be a N -closed set in Y. Now h −1 (F) = f −1 (F) ∩ g −1 (F), by elementary set theory. Since f is N * -continuous, f −1 (F) is N * -closed in A and therefore, N -closed in X. Similarly, g −1 (F) is N * -closed in B and therefore, N -closed in X. Thus their union h −1 (F) is N -closed in X. ✷

Conclusion
In this paper, we introduce a new venture to establish more topologies on a non empty set. Such efforts prompt us to blissfully convey that these concepts are also applicable in other areas of General topology, Fuzzy topology, intuitionistic topology, ideal topology so on and so forth. The course of human history, unmistakably shown and revealed to us that many great leaps of learning, discoveries, and understanding come from a source not so anticipated, and that in any field of sciences or humanities, and in particular in the field of basic researches often bear fruit well within hundred years, so to say. However, the more we come to grapple with and invest our time and energy to comprehend anything that is new, the better will we be, in order to handle and deal with the challenges and queries that keep facing us in the future, and come up with better results and findings.