Ratio Mathematica Volume 47, 2023 Door spaces on N- Topology Loyala Foresith Spencer J * Davamani Christober M† Abstract In this article, we explore the idea of door space on N-topological space. Here, we discuss which door spaces in this space are related with Nτ submaximal. The equivalent conditions shows how it con- nects a N-topological property. Also, we derive various door spaces using separation axioms and discuss the characteristics of such door spaces. We take a strong form of open set in N-topological space and introduce a new door space called Nτβ - door space. In addition, we analyze Nτβ-door space and discuss the relationship between a Nτβ-locally closed set and Nτ-closed set. Keywords: N-topology; door space; sub-maximal; Nτdβ-open sets. 2020 AMS subject classifications: 54A05, 54A10, 54C05 1 *The American College, Madurai, India; e-mail: spencerjraja@gmail.com. †The American College, Madurai, India; e-mail:christober.md@gmail.com. 1Received on September 15, 2022. Accepted on March 10, 2023, Published on April 4, 2023. DOI: 10.23755/rm.v39i0.957. ISSN: 1592-7415. eISSN: 2282-8214. ©The Authors. This paper is published under the CC-BY licence agreement. 367 Loyala Foresith Spencer J, Davamani Christober M 1 Introduction In 1955, J.L. Kelly [2], introduced the term ’door’ in classical topology. He investigated the relationship between the different topological spaces and door spaces. McCartan [5] found three types of door spaces and established the con- cepts of connected door spaces and maximally connected. Muckenhoupt and Williams [6] concluded that there exists a non-zero Borel measure in every con- nected door space. Mathew [4]enquired about hyper-connected door space and proposed its related concepts. In 1983, Monsef et al.[1] initiated the study of β-open sets and β-continuity in a topological space. This article explores the idea of door space in N - topological space. Here, we introduce Nτ-door space and discuss which door spaces are related with Nτ submaximal. The equivalent conditions shows how it connects with Nτ frontier, a N-topological property. Also, we derive various door spaces using separation axioms and discuss the characteristics of such door spaces. We take a strong form of open set in N-topological space and introduce a new door space called Nτβ - door space . Finally, we introduce Nτβ - locally closed sets and its properties which are all the essential tool for the future development of this concept. 2 Nτ - door spaces In this section, we establish various door spaces in N-topological space and explicit its properties. Moreover, discussed about the sub maximal concept and analyze its characterization in this space. Definition 2.1. A (P, Nτ) - space is called Hausdorff space of N topological space if for every given pair of different points z, q ∈ P ,there exists R, S ∈ Nτ O(P) such that z ∈ R, q ∈ S, R ∩ S = ϕ and is denoted by Nτ - Hausdorff space. Definition 2.2. A (P, Nτ) - space is called semi - hausdorff space of N topologi- cal space if for every given pair of different points z, q ∈ P ,there exists R, S ∈ Nτ SO(P) such that z ∈ R, q ∈ S, R∩S = ϕ and is denoted by Nτ - semi hausdorff space. Theorem 2.1. Each Nτ hausdorff space of (P, Nτ) is Nτ - semi hausdorff space. Proof. Suppose (P, Nτ) is a Nτ hausdorff space then there exisits two disjoint points which can be isolated by disjoint Nτ open set. Since each Nτ open set is a Nτ - semi open then we may conclude that each Nτ hausdorff space of (P, Nτ) is Nτ - semi hausdorff space. 368 Door spaces on N- Topology Remark 2.1. The inverse of the theorem 5.2.3 need not be true which is shown below. Let P = {l, m, n, o, p}. For N = 3, consider τ1 = {ϕ, {l}, {l, p}, {l, m, o, p}, P}, τ2 = {ϕ, {l, m, o}, {m, o}, P} and τ3 = {ϕ, {m, o, p}, {l, m, o, p}, P}. Then 3τ = {ϕ, {l}, {p}, {l, p}, {m, o}, {l, m, o}, {m, o, p}, {l, m, o, p}, P}. Here {l, n} is 3τ - semi open set but not 3τ - open. Definition 2.3. If every subset of (P, Nτ) is either Nτ - open or Nτ-closed then (P, Nτ) is called as Nτ - door space. Example 2.1. Let S = {x, y, z}. For N = 2, τ1 = {ϕ, {z}, {x, z}, S}. Then 2τ O(S) = {ϕ, {x}, {z}, {x, z}, S}, 2τ C(S) = {ϕ, {y, z}, {x, y}, {y}, S}. There- fore S is a 2τ - door space. Theorem 2.2. Every subspace G of a Nτ - door space is Nτ - door space. Proof. Let (P, Nτ) be a door space. Let G ⊆ P and U ∈ G. Since P is a Nτ door space then U is either Nτ open or Nτ closed in P and hence in G. Therefore G is also a Nτ door space. Definition 2.4. A subset S of (P, Nτ) is Nτ - dense if Nτ - cl(S) = P . Definition 2.5. A N-topological space (P, Nτ) is sub maximal if every Nτ - dense subset of P is NτO(P) and is denoted as Nτ- sub maximal. Theorem 2.3. Every door space (P, Nτ) is Nτ - sub maximal. Proof. Let (P, Nτ) be a door space and V ⊂ P be a Nτ dense. If V is not a Nτ open then it is Nτ closed since P is a Nτ door space. Now V = Nτcl(V ) = P and V is Nτ open. Hence P is Nτ - sub maximal. Theorem 2.4. Every subspace L of a sub maximal space (P, Nτ) is again a Nτ - sub maximal. Proof. Let A be a Nτ dense subset of L. Then Nτcl(A) ∩ L = L and so L ⊂ Nτcl(A). Since A∪(P −Nτcl(A)) is Nτ dense in P then it is an Nτ open subset of P . Hence S ∩ (A ∪ (P − Nτcl(A))) = A is Nτ - open in L or equivalently L is Nτ - submaximal. Theorem 2.5. In (P, Nτ), the following conditions are equivalent 1. P is Nτ - submaximal. 2. For any A ⊂ P , the subspace Nτ - Fr(A) = Nτ-cl(A)− Nτ - int(A) = Nτ-cl(A) ∩ Nτ-cl(P − A) is discrete. 369 Loyala Foresith Spencer J, Davamani Christober M Proof. (1) =⇒ (2) Let y ∈ NτFr(A). Since A is Nτ dense in Nτcl(A) then so is A ∪ {y}. Since Nτcl(A) is sub maximal according to theorem 5.2.11 then A ∪ {y} = Nτcl(A) ∩ U where U is Nτ open in P . In the same way it can seen that Ac ∪ {y} = Nτcl(Ac) ∩ V where V is Nτ open in P . Thus {y} = (A ∪ {y}) ∩ (Ac ∪ {y}) = Nτcl(A) ∩ Nτcl(P − A) ∩ U ∩ V . Hence {y} is Nτ open in NτFr(A) and so NτFr(A) is discrete. (2) =⇒ (1) Let A be a Nτ dense in P . By assumption, Nτcl(A)−Nτint(A) = P − Nτint(A) is discrete and thus A − Nτint(A) is its Nτ - open subset. Hence A − Nτint(A) = (P − Nτint(A)) ∪ U where U is Nτ open in P . Thus A − Nτint(A) ⊂ U and so A−Nτint(A) ⊂ U −Nτint(A). For the reverse inclusion if y ∈ U − Nτint(A) then y ∈ (P − Nτint(A)) ∩ U = A − Nτint(A). This shows that A − Nτint(A) = U − Nτint(A) and hence A = U ∪ Nτint(A). Thus A is Nτ open in P . Theorem 2.6. In (P, Nτ), the following conditions are equivalent 1. P is Nτ - submaximal. 2. Every Nτ - pre open subset of P is Nτ - open. Proof. (1) =⇒ (2) Let A be a Nτ pre open in P . Then A ⊂ Nτint(Nτcl(A)). Since A is Nτ dense in Nτcl(A) and Nτcl(A) is Nτ sub maximal accord- ing to theorem 5.2.11 then A is Nτ open in Nτcl(A). Thus A is Nτ open in Nτint(Nτcl(A)). Since Nτint(Nτcl(A)) is Nτ open in P then we may con- clude that A is Nτ open in P . (2) =⇒ (1) Let A be a Nτ dense in P . Since A ⊂ P = Nτint(P) = Nτint(Nτcl(P)) then A is Nτ - pre open and by assumption, Nτ open. This shows that P is Nτ - sub maximal. Theorem 2.7. If (P, Nτ) is submaximal and U ⊂ P then U is Nτ - open iff it is the intersection of a Nτ - dense and Nτ - regular open [3]. Proof. It is enough to prove that for every Nτ open set U, we have U = D ∩ V where D is Nτ dense and V is Nτ regular open since the reverse inclusion is trivial. Clearly U ⊂ Nτint(Nτcl(U)). Thus U = Nτcl(U) − (Nτcl(U) − U) = Nτcl(U) ∩ (P − (Nτcl(U) − U)) = Nτint(Nτcl(U)) ∩ (U ∪ P − Nτcl(U)) where Nτint(Nτcl(U)) = V is Nτ - regular open and U ∪ (P − Nτcl(U)) = D is Nτ - dense. Theorem 2.8. The homeomorphic image of Nτ - door space is a Nσ - door space. Proof. Consider (Z, Nτ) and (Q, Nσ) are door spaces and η : Z → Q be a home- omorphism. Let U ⊆ Q. Consider η−1(U) ⊆ Z. Since Z is Nτ - door space then η−1 is either Nτ - open or Nτ - closed in Z. Now η(η−1(U)) = U and U is either Nσ- open or Nσ - closed in Q. 370 Door spaces on N- Topology Definition 2.6. In (P, Nτ), if every subset of P is either Nτ semi-open or Nτ semi-closed then (P, Nτ) is semi-door space and is denoted by Nτ semi-door space Theorem 2.9. The homeomorphic image of Nτ semi-door space is a Nσ semi- door space. Proof. Let (Z, Nτ) and (Q, Nτ) are N - topological spaces and (Z, Nτ) be Nτ semi-door space. A mapping η : Z → Q be a homeomorphism. Let P ⊂ Q. Consider η−1(P) ⊂ Z, since Z is Nτ semi - door space then η−1(P) is either Nτ semi -open or Nτ semi - closed. So η(η−1(P)) = P is either Nτ - semi open or Nτ - semi closed. Hence Q is a Nτ - semi door space. Theorem 2.10. A Nτ - clopen subspace of a Nτ semi-door space is Nτ semi- door space. Proof. Let (Z, Nτ) be semi-door space. Let Q be a Nτ-clopen subset of Z. Let A ⊆ Q and A ⊆ Z. Since Z is a Nτ semi-door space, then A is either Nτ semi open or Nτ semi closed in Z. Since Q is Nτ open and Nτ closed, then A is either Nτ semi open or Nτ semi closed in Q. Hence Q is Nτ semi-door space. Theorem 2.11. If (P, Nτ) be a door space and if z ∈ P , S is a Nτ - neighbour- hood of z, then S − z ∈ Nτ and S ∈ Nτ. Proof. Let S be an neighbourhood of a point v and if v is Nτint(S) then it is enough to prove that S − v is Nτ open. If we assume S − v is not Nτ - open then P − (S − v) = (P − S) ∪ v should be Nτ open. This contradicts that v = S ∩ ((P − S) ∪ v) should be Nτ - open. Definition 2.7. In (P, Nτ), if every two disjoint points in P can be isolated by disjoint Nτ open sets then (P, Nτ) is a hausdorff door space and is denoted by Nτ-hausdorff door space. Definition 2.8. In (P, Nτ), if every two disjoint points in P can be isolated by disjoint Nτ semi open sets then (P, Nτ) is a semi - hausdorff door space and is denoted by Nτ semi-hausdorff door space. Definition 2.9. A Nτ - semi door space is said to be hausdorff semi-door space of (P, Nτ) if a given pair of different points r, s ∈ P , there exist M, N ∈ NτO(P) such that r ∈ M, s ∈ N, M ∩ N = ϕ and is denoted by Nτ-hausdorff semi door space. Proposition 2.1. Every Nτ - hausdorff door space is Nτ semi-hausdorff door space. 371 Loyala Foresith Spencer J, Davamani Christober M Proposition 2.2. If (P, Nτ) be hausdorff door space and z ∈ P then r ∈ Nτ SO(P) ⇐⇒ r ∈ Nτ. Proposition 2.3. If (P, Nτ) be hausdorff door space and Z, Q ∈ (P, Nτ). If z ∈ NτSO(P) and Q ∈ Nτ then Z ∩ Q ∈ Nτ SO(P). Theorem 2.12. Nτ - semi-hausdorff door space has atmost one limit point. Proof. Consider (P, Nτ) be a hausdorff space. Let a, b are distinct limit points in P. Since P is Nτ - semi-hausdorff, ∃ G, H ∈ SO(P) : a ∈ G, b ∈ H and G ∩ H = ϕ. Since P is Nτ - door space then U = {G − {a}} ∪ {b} is either Nτ - open or Nτ - closed. Suppose if it is Nτ - open then by theorem 5.2.19 U ∩ H = {b} is Nτ - semi open and hence by theorem 5.2.20 U ∩ H is Nτ - open. Otherwise Uc is Nτ - open and Uc ∩ G = {a} is Nτ - semi open and Nτ - open by propositions 5.2.24 and 5.2.25. Hence at least one of the two point will be isolated in P and by contradiction the result is proved. Proposition 2.4. A Nτ - hausdorff semi-door space has atmost one limit point. Proof. Proof is similar as discussed in the previous result. 3 Nτβ-door space and locally Nτβ-closed set In this section, we introduce and analyze Nτβ - door space and Nτβ - locally closed sets. Definition 3.1. A subset U of P is said to be locally Nτ-closed set if U = R ∩ S where R is a Nτ - closed in P . The set of all locally Nτ - closed sets are denoted by LNτ-Cl(P) Definition 3.2. A subset U of P is said to be locally Nτβ-closed set if U = R∩S where R is a Nτβ - open and S is Nτβ-closed in P . The set of all locally Nτβ - closed sets are denoted by LNτβ-Cl(P) Definition 3.3. If every subset of (P, Nτ) is either Nτβ-open or Nτβ - closed then (P, Nτ) is called Nτβ - door space and signified by Nτβd. Example 3.1. Let S = {x, y, z}. For N = 2, τ1 = {ϕ, S}, τ2 = {ϕ, {x, y}, S}. Then 2τ O(S) = {ϕ, {x, y}, S}, 2τCl(S) = {ϕ, {z}, S}. Here S is a 2τβ - door space. LNτ-Cl(S) = {ϕ, {x, y}, {z}, S} and LNτβ-Cl(P) = {ϕ, {x}, {y}, {z}, {x, y}, {y, z}, {z, x}, S}. Remark 3.1. From the example 5.3.4, we get 372 Door spaces on N- Topology 1. Every locally Nτ-closed set is locally Nτβ- closed set but its converse need not be true. 2. Every Nτ-door space is a Nτβd but its converse need not be true. Remark 3.2. If (P, Nτ) is Nτβ - door space then LNτβ-Cl(P) = ℘(P). Theorem 3.1. Let G ⊆ P . Then the following are equivalent: 1. G is locally Nτβ - closed sets 2. G = F ∩ Nτ-cl(G) for some Nτβ-open set F . Proof. (1) =⇒ (2) Let G ∈ LNτβcl(P). Now there will be a Nτβ - open set F and a Nτβ - closed subset E such that G = F ∩ E since G ⊆ F and G ⊆ Nτβcl(P) then Nτβcl(P) ⊆ E. Hence F ∩ Nτβcl(P) ⊆ F ∩ E = G. Therefore G = F∩ Nτβcl(G). This proves (1) =⇒ (2) . (2) =⇒ (1) By definition, Nτβcl(G) is Nτβ - closed. ∴ G = F∩LNτβcl(G) ∈ NτβLC(P). 4 Conclusion In this paper, we introduced the idea of door spaces in N-topological space. Their structural properties have been discussed and emphasized. Some of the im- portant results arrived through illustrated examples. The importance is to analyse the relationship with other N- topological properties. So, We investigated sub- maximal concepts and locally closed sets through N-topology. In addition, we introduce Nτβ-door space and discuss the relationship between a Nτβ-locally closed set and Nτ-closed set. With the help of these locally closed sets it can be extend to introduce locally continuous maps in this topological space. These prime ideas can open the future scope of this concept and extended to other re- search areas of topology such as Fuzzy topology, Digital topology, and so on. References [1] Abd el-monsef, M.E., El-deeb S.N. and Mahmoud R.A. β-open sets and β- continuous mappings, Bull. Fac. Sci., 12:77 – 90,1983. [2] Kelly J.L. General topology, Princeton, NJ. D. Van Nastrand, 1955. [3] Loyala F. Spencer J. and Davamani Christober M. Theta open sets in N- topology, Ratio Mathematica, 43:163 – 175, 2022. 373 Loyala Foresith Spencer J, Davamani Christober M [4] Mathew P.M. On hyper connected spaces, Indian j. pure appl. math..1988. [5] McCartan S.D. Door Spaces are identifiable, Proceedingds of Roy irish acad. sect. A, 87 (1): 13 – 16, 1987. [6] Muckenhoupt B. and Williams V. Borel measures on connected Door Spaces, Riv. Mat. Univ. Parma., 3(2): 103 – 108, 1973. 374