Ratio Mathematica Volume 41, 2021, pp. 283-290 Contra Nα-I-Continuity over Nano Ideals S. Vijaya* P. Santhi† A. Yuvarani‡ Abstract The conceptualization of Nα-I-open sets and Nα-I-continuous func- tions in nano ideal topology are used to study contra Nα-I-continuity. Also the characteristics and behaviours of contra Nα-I-continuity based on Nano Urysohn Space and Nano Ultra Hausdorff Space are dis- cussed. Keywords: CNα-Cts function, CNα-I-Cts function, Nα-I-T2 space, Nα-I-connected. 2020 AMS subject classifications: 54A05, 54B05. 1 *Thiagarajar College, Madurai, Tamil Nadu, India; viviphd.11@gmail.com. †The Standard Fireworks Rajaratnam College for Women, Sivakasi, Tamil Nadu, India; saayphd.11@gmail.com. ‡The American College, Madurai, Tamil Nadu, India; yuvamaths2003@gmail.com. 1Received on August 7, 2021. Accepted on November 30, 2021. Published on December 31, 2021. doi: 10.23755/rm.v41i0.651. ISSN: 1592-7415. eISSN: 2282-8214. ©Vijaya et al. This paper is published under the CC-BY licence agreement. 283 S. Vijaya, P. Santhi, A. Yuvarani 1 Introduction The ideal concept in topology was developed by Kuratowski [Kuratowski, 1966].The notion of α-I-continuity was introduced in 2004 [A. Acikgoz and Yuk- sel, 2004].The conception of nano topology was initated by L.Thivagar [Thivagar and Richard, 2013a].In addition to that the concept of continuity, α-continuity, kernal and clopen in Nano topology was introduced by [Karthiksankari and Sub- bulakshmi, 2019] [Thivagar and Richard, 2013b] and [M. Lellis Thivagar and SuthaDevi, 2017].Parimala and Jafari [Parimala and Jafari, 2018] had worked on Nano ideals.This work aims the introduction of contra Nα-I-continuous func- tions by applying the concept of Nα-I-open and Nα-I-continuity in nano ideal topology.Also this contra Nα-I-continuity are compared with some existing func- tions.Moreover, new class of functions are obtained. At every places the new notions have been substantiated with suitable examples.Throughout this article we use the notation NTS, NITS, N-regular, N-open, Nα-open, N-clopen, Nα-Cts for Nano Topological Spaces, Nano Ideal Topological Spaces, nano regular space, nano open, nano α-open, nano clopen, nano α-continuous respectively.Similar notations are used for their respective closed sets. 2 Preliminaries Definition 2.1. [M. Lellis Thivagar and SuthaDevi, 2017] Let (U, τR(X)) be a NTS and S is a subset of U.The nano kernel of S is defined as NKer(S)=∩{U : S is a subset of U, U ∈ τR(X)}. Theorem 2.1. [M. Lellis Thivagar and SuthaDevi, 2017] Let (U, τR(X)) be a NTS and A1, A2 ⊆ U.We have 1. x ∈ NKer(A1) iff for any N-closed set F containing x, A1 and F are disjoint, 2. If A1 ⊆ NKer(A1) and then A1= Ker(A1) if A1 is N-open in U, 3. If A1 ⊆ A2, then NKer(A1) ⊆ NKer(A2). Definition 2.2. [Thivagar and SuthaDevi, 2016] A NTS (U, τR(X)) along with an ideal I defined on U is called as a NITS and is denoted by (U, τR(X),I).Throughout this paper U represents a NTS (U, τR(X)) and UI represents a NITS (U, τR(X),I). Definition 2.3. [Rajasekaran and Nethaji, 2018] Let (U, τR(X),I) be a nano ideal topological space and A ⊆ U.Then A is said to be Nα-I-open if A ⊆ Nint(Ncl∗(Nint (A))).The complements of Nα-I-open is Nα-I-closed set. 284 Contra Nα-I-Continuity over Nano Ideals Theorem 2.2. [V. Inthumathi and Krishnaprakash, 2020] Let (U1, τR(X1),I) be a NITS and (U2, τR(X1)) be a NTS.Then h : U1 → U2 is called Nα-I-Cts on U1 if h−1(S) is Nα-I-open in U1 for any N-open set S in U2. Definition 2.4. Bhuvaneswari and Nagaveni [2018] A NTS (U, τR(X)) is called N-regular Space, if for each N-closed set T and each point x 6∈ T, ∃ disjoint N-open sets G and H such that x ∈ G and T ⊂ G. 3 Contra Nα-I-Continuity The notations used are CNα-open, CN-Cts function, CNα-Cts function, CNα- I-Cts function for contra nano α-open, contra nano continuous, contra Nα- con- tinuous, contra Nα-I-continuous function resp. Definition 3.1. Let (U1, τR(X1)) and (U2, τR′ (X2)) be NTS.Then h : U1 → U2 is CNα-Cts if h−1(S) is Nα-closed in U1 whenever S is N-open set in U2. Definition 3.2. Let h : (U1, τR(X1),I) → (U2, τR′ (X2)) is CNα-I-Cts if h−1(S) is Nα-I-closed in U1 whenever S is N-open set in U2. Example 3.1. Let U1={i,j,k,l}, U1/R={{i},{j},{k},{l}} and X1={i}.Then τR(X1) ={U1,φ,{i}}.Let I={φ}.Here the N α-I-open sets are{U1,φ,{i},{i,j},{i,k},{i,l},{i, j,k},{i,j,l},{i,k,l}}. Let U2={m,n,o,p} with U2/R′={{m},{n},{o,p}} and X2={n, o}.Then τR′ (X2)={ U2,φ,{n},{o,p},{n,o,p}}.We define h:(U1,τR(X1),I)→( U2, τR′ (X2)) as f(i)=m, f(j)=n, f(k)=o and f(l)=p.Then h−1(S) is Nα-I-closed in U1 when- ever S is N-open in U2.Therefore h is CNα-I-Cts. Proposition 3.1. 1. Any CNα-I-Cts function is CNα-Cts. 2. Any CN-Cts function is CNα-I-Cts. Proof. (i) Let h : (U1, τR(X1),I) → (U2, τR′ (X2)) be a CNα-I-Cts function.Let S be a N-open in U2.Since h is CNα-I-Cts, h−1(S) is Nα-I-closed in U1.We know that each Nα-I-closed set is Nα-closed.Hence h−1(S) is Nα-closed in U1.Hence h is CNα-Cts. (ii) Let h : (U1, τR(X1),I) → (U2, τR′ (X2)) be a CN-Cts function.Let S be a N- open set in U2.Since h is CN-Cts, h−1(S) is N-closed in U1.It is obvious that every N-closed set is Nα-I-closed.Thus h−1(S) is Nα-I-closed in U1.Which implies h is CNα-I-Cts function.2 Example 3.2. CNα-Cts ; CNα-I-Cts Let U1={i,j,k,l} with U1/R={{i},{j,k},{l}} and X1={l}.Then τR(X1)={U1,φ,{l}}. Let I={φ,{l}}.Here the Nα-open sets are {U1,φ,{l},{i,l},{j,l},{k,l},{i,j,l},{i,k,l}, 285 S. Vijaya, P. Santhi, A. Yuvarani {j,k,l}} and Nα-I-open sets are {U1,φ,{l}}.Let U2={m,n,o,p} with U2/R′={{m}, {n,o},{p}} and X2={m,n}.Then τR′ (X2) = {U2,φ,{m},{n,o},{m,n,o}}. We define h : (U1, τR(X1),I) → (U2, τR′ (X2)) as h(i)=m, h(j)=n, h(k)=o and h(l)=p.Then h−1(S) is Nα-closed in U1 but not Nα-I-closed whenever S is N-open set in U2. Hence h is CNα-Cts but not CNα-I-Cts function. Example 3.3. CNα-I-Cts ; CN-Cts Let U1={i,j,k,l} with U1/R= {{i},{j},{k},{l}} and X1={i}.Then τR(X1)={U1,φ,{ i}}.Let I = {φ}.Here the Nα-I-open sets are {U1,φ,{i},{i,j},{i,k},{i,l},{i,j,k},{i,j, l},{i,k,l}}.Let U2={m,n,o,p} with U2/R′={{m},{o,p},{n}} and X2={n,o}.Then τR′ (X2) = {U2,φ,{n},{o,p},{n,o,p}}.We define h : (U1, τR(X1),I) → (U2, τR′ (X2)) as h(i) = m, h(j) = n, h(k) = o and h(l) = p.Then h−1(S) is Nα-I-closed in U1 but not N-closed whenever S is N-open set in U2. Hence h is CNα-I-Cts but not CN-Cts function. Theorem 3.1. Let h : (U1, τR(X1),I) → (U2, τR′ (X2)), then the following state- ments are equivalent: 1. h is CNα-I-Cts, 2. for each N-closed subset T of U2, h−1(T) ∈ NαIO(U1), 3. for each x ∈ U1 and each N-closed set T of U2 containing h(x), ∃ U ∈ NαIO(U1) such that h(U) ⊂ T, 4. h(NαI-cl(V)) ⊂ NKer(h(V)) for each V ⊆ U1, 5. NαI-cl(h−1(W)) ⊂ h−1(NKer(W)) for each W ⊆ U2. Proof. (i) ⇒ (ii) and (ii) ⇒ (iii) are obvious. (iii) ⇒ (ii) Let T be any N-closed set of U2 and x ∈ h−1(T).Then h(x) ∈ T and ∃ Ux ∈ NαIO(U1) such that h(Ux) ⊂ T.Therefore, we obtain h−1(T) = ∪{ Ux : x ∈ h−1(T)} and hence h−1(T) ∈ NαIO(U1). (ii) ⇒ (iv)Let V ⊆ U1.If y 6∈ NKer(h(V)), then by Thm 2.1, ∃ a N-closed set T of U2 containing y such that h(V) ∩ T=φ.Therefore V ∩ h−1(T) = φ and NαI-cl(V) ∩ h−1(T)=φ.Hence h(NαI-cl(V)) ∩ T=φ and y 6∈ h(NαI-cl(V)).Thus h(NαI-cl(V)) ⊂ NKer(h(V)). (iv) ⇒ (v) Let W ⊆ U2.By the hypothesis and Thm 2.1, h(NαI-cl(h−1(W))) ⊂ NKer(h(h−1(W))) ⊂ NKer(W) and NαI-cl(h−1(W)) ⊂ h−1(NKer(W)). (v) ⇒ (i) Let W be a N-open set of U2.By Thm 2.1, NαI-cl(h−1(W)) ⊂ h−1(NKer(W)) = h−1(W) and NαI-cl(h−1(W)) = h−1(W).Therefore h−1(W) is Nα-I-closed in (U1, τR(X),I).2 286 Contra Nα-I-Continuity over Nano Ideals Theorem 3.2. If a function h : (U1, τR(X1),I) → (U2, τR′ (X2)) is CNα-I-Cts and V is N-regular, then h is Nα-I-Cts. Proof. Let x ∈ U1 and Y a N-open set of U2 containing h(x).Since U2 is N-regular, ∃ a N-open set Z in U2 containing h(x) such that Ncl(Z) ⊂ Y.Since h is CNα-I-Cts, by the above theorem, ∃ X ∈ NαIO(U1) such that h(X) ⊂ Ncl(Z).Therefore h(X) ⊂ Ncl(Z) ⊂ Y.Hence h is Nα-I-Cts.2 Definition 3.3. A function h : (U1, τR(X1),I) → (U2, τR′ (X2)) satisfy the Nα-I- interiority rule if NαI-int(h−1(Ncl(W))) ⊂ h−1(W) Whenever W is N-open set of (U2, τR′ (X2)). Theorem 3.3. If a function h : (U1, τR(X1),I) and (U2, τR′ (X2)) is CNα-I-Cts and satisfies Nα-I-interiority rule, then h is Nα-I-Cts. Proof. Let Y be any N-open set of U2. Since h is CNα-I-Cts and Ncl(Y) is N-closed, by Thm 3.1, h−1(Ncl(Y)) is Nα-I-open in (U1, τR(X),I).By hypothesis of h, h−1(Y) ⊂ h−1(Ncl(Y)) ⊂ NαI-int(h−1(Ncl(Y))) ⊂ NαI-int(h−1(Y)) ⊂ h−1(Y).Thus, we ob- tain h−1(Y)=NαI-int(h−1(Y)) and consequently h−1(Y) ∈ NαIO(U).Therefore h is Nα-I-Cts.2 Theorem 3.4. Let (U1, τR(X1),I) be any NITS and h : (U1, τR(X1),I) → (U2, τR′ (X2)) be a function and g : U1 → U1 × U2 be the graph function, given by g(x) = (x, h(x)) for every x ∈ U1.Then f is CNα-I-Cts if and only if g is Nα-I-Cts. Proof. Let x ∈ U1 and let T be a N-closed subset of U1 × U2 containing g(x).Then T ∩ ({x}× U2) is N-closed in {x}× U2 containing g(x).Also {x}× U2 is home- omorphic to U2. Hence {y ∈ U2 : (x, y) ∈ T} is a N-closed subset of U2.Since h is CNα-I-Cts, ∪ { h−1(Y) ∈ U2 : (x, y) ∈ T } is a Nα-I-open subset of (U1, τR(X1),I).Further, x ∈∪ { h−1(Y) ∈ U2 : (x, y) ∈ T }⊂ g−1(T).Hence g−1(T) is Nα-I-open.Then g is CNα-I-Cts. Conversely, let F be a N-closed subset of U2.Then U1 × F is a N-closed subset of U1 × U2.Since g is CNα-I-Cts, g−1(U1 × F) is a Nα-I-open subset of U1. Also, g−1(U1 × F)=h−1(F).Hence h is CNα-I-Cts.2 Definition 3.4. A NITS (U1, τR(X1),I) is called Nα-I-T2 if for any distinct two points x, y ∈ U1, ∃ X, Y ∈ NαIO(U1) containing x and y, resp., such that X ∩ Y=φ. Definition 3.5. 1. A NTS (U1, τR(X1)) is termed as a N-Urysohn Space if for any two distinct points x, y ∈ U1, ∃ disjoint N-open subsets x ∈ A, y ∈ B such that the N-closures A and B are disjoint N-closed subsets of U1. 2. A NTS (U1, τR(X1)) is called N-Ultra Hausdorff if any two distinct points of U1 can be separated by disjoint N-clopen sets. Theorem 3.5. If (U1, τR(X1),I) is an NITS and for any two distinct points x1 , x2 ∈ U1, ∃ a function h into a N-Urysohn Space (U2, τR′ (X2)) such that h(x1) 6= h(x2) and h is CNα-I-Cts at x1 , x2, then (U1, τR(X1),I) is Nα-I-T2. 287 S. Vijaya, P. Santhi, A. Yuvarani Proof. Let x1 , x2 be any two distinct points of U1.Then by hypothesis there is a N- Urysohn Space (U2, τR′ (X2)) and a function h : (U1, τR(X1),I) and (U2, τR′ (X2)), which satisfies the required condition.Let yi= h(xi) for i=1,2.Then y1 6= y2. Since (U2, τR′ (X2)) is N-Urysohn, ∃ N-open neighbourhoods Xy1 and Xy2 of y1, y2 respectively in U2 such that Ncl(Xy1 ) ∩ Ncl(Xy2 )=φ.Since h is CNα-I-Cts at xi, ∃ Nα-I-open neighbourhoods Wxi of xi in U1 such that h(Wxi ) ⊂ Ncl(Xyi ) for i=1,2. Hence we get Wx1 ∩ Wx2 =φ because Ncl(Xy1 ) ∩ Ncl(Xy2 )=φ.Therefore (U1, τR(X1),I) is Nα-I-T2.2 Corolary 3.1. If h is a CNα-I-Cts injective function of a NITS (U1, τR(X1),I) into a N-Urysohn space (U2, τR′ (X2)), then (U1, τR(X1),I) is a Nα-I-T2 space. Proof. For any to two distinct points x1 , x2 in U1, h is CNα-I-Cts function of U1 into a N-Urysohn space (U2, τR′ (X2)) such that h(x1) 6= h(x2) because h is injective.By Thm 3.5, the space (U1, τR(X1),I) is Nα-I-T2.2 Theorem 3.6. If h is a CNα-I-Cts injective function of a NTS (U1, τR(X1),I) into N-Ultra Hausdorff space (U2, τR′ (X2)), then (U1, τR(X1),I) is a Nα-I-T2 space. Proof. Let the pair of distinct points of U1 be x1 , x2.Since f is injective, U2 is N-Ultra Hausdorff h(x1) 6= h(x2) ∃ N-clopen sets Z1, Z2 such that h(x1) ∈ Z1, h(x2) ∈ Z2 and Z1 ∩ Z2=φ.Then xi ∈ h−1(Zi) ∈ NαIO(U1) for i=1,2 and h−1(Z1) ∩ h−1(Z2)=φ.Therefore (U1, τR(X),I) is a Nα-I-T2 space.2 Definition 3.6. Let h : (U1, τR(X1),I) → (U2, τR′ (X2)).The graph G(h) of the function h is called be CNα-I-closed in U1 × U2 if for any (x1, x2) ∈ (U1 × U2)\G(h), ∃ A ∈ NαIO(U1) and a N-closed set T of U2 containing x2 such that (U1 × U2) ∩ G(h)=φ. Lemma 3.1. Let h : (U1, τR(X1),I) → (U2, τR′ (X2)).The graph G(h) of the function h is CNα-I-closed in U1 × U2 if and only if for each (x1, x2) ∈ (U1 × U2)\G(h), ∃ A ∈ NαIO(U1, x1) such that h(A) ∩ Ncl(T)=φ where T is a N-closed subset of U1 × U2 containing g(x1). Theorem 3.7. If h : (U1, τR(X1),I) → (U2, τR′ (X2)) is a CNα-I-Cts function and U2 is a N-Urysohn space, then G(h) is CNα-I-closed in U1 × U2. Proof. Let (x1, x2) ∈ (U1 × U2)\G(h).Then x2 6= h(x1) and ∃ N-open set A, B of U2 such that h(x1) ∈ A, x2 ∈ B and Ncl(A) ∩ Ncl(B)=φ.Since h is CNα-I-continuous, ∃ U ∈ NαIO(U1,x1) such that h(U) ⊂ Ncl(A).Therefore h(U) ∩ Ncl(B) =φ.Hence G(h) is CNα-I-closed.2 Theorem 3.8. If h : (U1, τR(X1),I) → (U2, τR′ (X2)) is a CNα-I-Cts function and (U2, τR′ (X2)) is T2, then G(h) is CNα-I-closed. Proof. Let (x1, x2) ∈ (U1 × U2)\G(h).Then x2 6= h(x1) and ∃ N-open set B of U2 such that h(x1) ∈ B, x2 6= B. Since h is CNα-I-Cts, ∃ U ∈ NαIO(U1,x1) such that 288 Contra Nα-I-Continuity over Nano Ideals h(U) ⊂ Ncl(B).Therefore h(U) ∩ (U2 - B)=φ and U2-B is a N-closed set of U2 containing x2.Hence G(h) is CNα-I-closed.2 Definition 3.7. A NITS (U, τR(X),I) is called Nα-I-connected if there exists Nα-I- open sets A and B which form a separation of X. Proposition 3.2. A CNα-I-Cts image of a Nα-I-connected space is connected. Definition 3.8. A NITS (U, τR(X),I) is called Nα-I-normal if given any non-empty disjoint N-closed sets T and F such that ∃ Nα-I-open sets A of T and B of F such that A ∩ B=φ. Definition 3.9. A NTS (U, τR(X)) is called N-Ultra normal if given any non-empty disjoint N-closed sets T and F such that ∃ N-clopen sets A of T and B of F such that A ∩ B=φ. Theorem 3.9. If h : (U1, τR(X1),I) → (U2, τR′ (X2)) is a CNα-I-Cts closed injective function and (U2, τR′ (X2)) is N-Ultra-normal space, then (U1, τR(X1),I) is a Nα- I-normal space. Proof. Let the two disjoint N-closed subsets of U1 be F1 and F2. Since h is N- closed and injective, h(F1) ∩ h(F2)=φ where h(F1) and h(F2) are N-closed subsets of U2.Since U2 is N-Ultra normal, ∃ N-clopen sets Y1 of h(F1) and Y2 of h(F2) in U2 such that Y1 ∩ Y2=φ.Hence Fi ⊂ f−1(Yi), f−1(Yi) ∈ NαIO(U) for i=1,2 and f−1(Y1) ∩ f−1(Y2)=φ.Therefore (U1, τR(X),I) is a Nα-I-normal.2 Theorem 3.10. For the functions h : (U1, τR(X1),I) → (U2, τR′ (X2)) and g : (U2, τR′ (X2),I’) → (U3, τR′′ (X3)), We have 1. g ◦ h is Nα-I-Cts, if h is CNα-I-Cts and g is CN-Cts. 2. g ◦ h is CNα-I-Cts, if h is CNα-I-Cts and g is N-Cts. Remark 3.1. In general, g ◦ h is not CNα-I-Cts functions if g and f are CNα-I-Cts functions.The below example illustrate this result. Example 3.4. Let U1={i,j,k,l} with U1/R={{i,k},{j},{l}}, and X1={i,l}.Then τR(X1)={U1,φ,{l},{i,k},{i,k,l}}.Let I1={φ,j}.Let U2={m,n,o,p} with U2/R′={{m, n},{o,p}} and Y={o,p}.Then τR′ (X2)={U2,φ,{p},{m,o},{m,o,p}}.Let I2={φ,m}. Let W={t,u,v,w} with W/R′′={{t},{u,v},{w}} and Z={w}.Then τR′′ (Z)={W,φ,{w} }.Define h : (U1, τR(X),I1) → (U2, τR′ (X2)) by h(i)=n, h(j)=p, h(k)=m, h(l)=o and g : (U2, τR′ (Y),I2) → (U3, τR′′ (Z)) by g(m)=w, g(n)=t, g(o)=u, g(p)=v.Then h and g are CNα-I-Cts functions but (g ◦ h)−1(w)=k which does not belongs to Nα-I- closed in (U1, τR(X1),I). 289 S. Vijaya, P. Santhi, A. Yuvarani 4 Conclusion Through the above discussions we have summarized the conceptulation of contra Nα-I-continuity and its characteristics based on Nano Urysohn Space and Nano Ultra Hausdorff Space.Also, We compared contra Nα-I-continuity with some existing functions using suitable examples.Further, this concept may be ex- tended to Frechet Urysohn Space and Completely Hausdorff space in Nano Ideal Topology. References T. Noiri A. Acikgoz and S. Yuksel. On α-i-continuous and α-i-open functions. Acta Mathematica Hungarica, 105:27–37, 2004. M. Bhuvaneswari and N. Nagaveni. On nwg-normal and nwg-regular spaces. 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