EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 12, No. 4, 2019, 1553-1566 ISSN 1307-5543 – www.ejpam.com Published by New York Business Global Functions on n-generalized Topological Spaces Cherry Mae R. Balingit1,∗, Julius V. Benitez2 1 Department of Mathematics, College of Arts and Sciences, Central Mindanao University, Maramag, Bukidnon, Philippines 2 Department of Mathematics and Statistics, College of Science and Mathematics, Mindanao State University-Iligan Institute of Technology, Iligan City, Philippines Abstract. An n-generalized topological (n-GT) space is a pair (X,G ) of a nonempty set X and a collection G of n (n ∈ N) distinct generalized topologies (in the sense of A. Császár [1]) on the set X. In this paper, we look into G -continuous maps, G -open and G -closed maps, as well as G -homeomorphisms in terms of n-GT spaces and establish some of their basic properties and relationships. Moreover, these notions are also examined with respect to the component generalized topologies of the underlying spaces by defining and characterizing pairwise versions of the said types of mappings. 2010 Mathematics Subject Classifications: 54A05, 54C05, 54C08, 54C10, 54H99 Key Words and Phrases: G -continuous map, G -open map, G -homeomorphism 1. Introduction Open sets play as a fundamental notion that underlies almost any other topological concept. When A. Császár [1] in 2002 weakened the conditions that are to be satisfied by an open set, the landscape of topological spaces widened significantly. In [1], he defined a generalized topological space (briefly, GT space) (X,µ) as a pair of a nonempty set X and an associated family µ of subsets of X satisfying only the conditions that ∅ ∈ µ and an arbitrary union of sets in µ belongs to µ. Naturally, the elements of µ are termed µ-open sets. In the same space, the µ-closure cµ(A) of a subset A of X is also defined as the intersection of all µ-closed sets containing A while the µ-interior iµ(A) of A is the union of all µ-open sets contained in A [1]. Other basic properties of a GT space were cited in [6] and [5]. Following this generalization, the idea of utilizing two or more GTs to form a new type of topological space were deeply explored in many succeeding studies. Some of these results are seen in [5], [8], [9], [10], [12], [13], [14], [15], [16], [17], and [19]. In particular, ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v12i4.3502 Email addresses: cmrbalingit@gmail.com (C. Balingit), julius.benitez@g.msuiit.edu.ph (J. Benitez) http://www.ejpam.com 1553 c© 2019 EJPAM All rights reserved. C. Balingit, J. Benitez / Eur. J. Pure Appl. Math, 12 (4) (2019), 1553-1566 1554 an n-generalized topological space (briefly, n-GT space) is defined in [2] as a pair (X,G ), where X is a nonempty set and G = {µ1, . . . , µn} is a finite collection of n GTs on X, for some n ∈ N. In this space, the G -closure of A ⊆ X, denoted by cG (A), is the intersection of all sets F containing A where F = X \G for some G ∈ ∪nj=1µj while its G -interior, denoted by iG (A), is the union of all sets G contained in A for some G ∈ ∪nj=1µj . Conse- quently, A ⊆ X is said to be G -closed if cG (A) = A and is G -open if iG (A) = A. These mentioned sets in an n-GT space naturally satisfy basic properties analogous to that of a GT. Most relevant in this paper are that the arbitrary union G -open sets is also G -open and the arbitrary intersection of G -closed sets is also G -closed. Also, each µj-open set is G -open for all j = 1, . . . , n so that for any G -open set A, we can always find a j and a µj-open set Uj such that Uj ⊆ A. Additionally, by definition of G -interior and G -closure, for each A ⊆ X, we have iG (A) ⊆ A and A ⊆ cG (A). On a finer note, it is observed that iG (A) = n⋃ k=1 iµk(A) and cG (A) = n⋂ k=1 cµk(A). Functions that arise from these new spaces were also studied in detail. They were modified to fit the definition of open sets in each of these domains. Among the popular types of functions tackled are the continuous maps, open and closed maps, as well as homeomorphisms (see [3], [4], [6] and [11], [18]). This paper intends to examine new variations of continuity, open maps and homeomorphisms in terms of n-GT spaces and to inquire into the idea of localizing these mappings to the component GTs of the underlying spaces. 2. G -continuous Functions We first define and establish the existence of G -continuous maps: Definition 2.1. Let (X,GX) and (Y,GY ) be m-GT and n-GT spaces, respectively, where GX = {µ1, . . . , µm} and GY = {ν1, . . . , νn} for some m,n ∈ N. A function f : X → Y is said to be G -continuous at a point x ∈ X if for each GY -open set V containing f(x), there exists a GX-open set U containing x such that f(U) ⊆ V . The function f : X → Y is G -continuous if it is continuous at all points x ∈ X. Example 2.2. Let m,n ∈ N where m ≤ n and consider the m-GT space (X,GX) where X = [0,m] × R+ 0 and GX = {µ1, . . . , µm} such that µj = {∅} ∪ {Rjs : s ≥ 0} with Rjs = {(x, y) : j − 1 ≤ x ≤ j, y ≥ s}, and the n-GT space (Y,GY ) where Y = [0, n] ⊆ R and GY = {ν1, . . . , νn} such that νk = {∅} ∪ {[0, t] : t = 1, . . . , k}. Define f : X → Y such that (a, b) 7→ a. For each (a, b) ∈ X, there is a j0 such that j0− 1 < a ≤ j0 and, therefore, (a, b) ∈ Rj00 . Now, the GY -open sets containing f(a, b) = a are precisely the intervals [0, t], where j0 ≤ t ≤ n. Now, for each t, f(Rj00 ) = [j0 − 1, j0] ⊆ [0, t]. This means that f is G -continuous at (a, b) and because (a, b) ∈ X is arbitrary, f is therefore G -continuous. This type of mapping also manifests for vertex and arc sets of a directed graph as in the following: C. Balingit, J. Benitez / Eur. J. Pure Appl. Math, 12 (4) (2019), 1553-1566 1555 Example 2.3. Let D be a directed graph with no loops and multiple edges and with finite vertex set V (D) and arc set E(D) and let P1, . . . , Pn be the distinct maximal nontrivial paths in D. For each j = 1, . . . , n, let µj be a family of subsets of V (D) containing precisely the empty set and all subsets of V (D) whose elements induce a union of subpaths of the maximal path Pj . Then each µj is a GT on V (D) and consequently, if G = {µ1, . . . , µn}, then (V (D),G ) is an n-GT space. Moreover, for each k = 1, . . . , n, let Ek be the collection of arcs in the maximal path Pk and νk = P(Ek) be the power set of Ek. Clearly, each νk is a GT on E(D) and so if we put GE = {ν1, . . . , νn}, then (E(D),GE) is also an n-GT space. Define ϕ : E(D)→ V (D) such that for e = uv ∈ E(G), ϕ(e) = u. If U is a GV -open set containing ϕ(e) = u, then we can find a k and a Gk ⊆ U such that u ∈ Gk ∈ µk. Now, the vertices in Gk form a union of subpaths of the maximal path Pk and with arcs from Ek. Let E∗k be the collection of arcs formed by the vertices in Gk. By the definition of GE , E∗k ∈ νk and so it is GE-open, with ϕ(E∗k) ⊆ Gk ⊆ U . This implies that ϕ is G -continuous on e ∈ E(D). By the arbitrary nature of e, ϕ is G -continuous. In the succeeding discussions, we assume that (X,GX) and (Y,GY ) are m-GT and n-GT spaces, respectively, where GX = {µ1, . . . , µm} and GY = {ν1, . . . , νn} for some m,n ∈ N and adapt the notations U = m⋃ j=1 µj and V = n⋃ k=1 νk. For convenience, we also adapt the notations cX(A) and iX(A) to signify the GX -closure and GX -interior whenever A ⊆ X. Theorem 2.4. A function f : X → Y is G -continuous if and only if f−1(V ) is GX-open for every GY -open set V . Proof. If V is a GY -open set and x ∈ f−1(V ), then f(x) ∈ V and since f is G -continuous, we can find a GX -open set Ux containing x such that f(Ux) ⊆ V . Consequently, Ux ⊆ f−1(V ) and so f−1(V ) ⊆ ⋃ x∈f−1(V ) Ux ⊆ f−1(V ). Hence, f−1(V ) = ⋃ x∈f−1(V ) Ux is GX -open. Conversely, if x ∈ X and V is a GY -open set containing f(x), then U = f−1(V ) is a GX - open set containing x such that f(U) = V . Since x is arbitrary, we see that f is continuous. � A G -continuous map also satisfy the following properties which are analogous to that of a continuous map as in [7]: Theorem 2.5. Let f : X → Y be a map. Then the following statements are equivalent: 1. f is G -continuous; 2. f−1(B) is GX-closed for each GY -closed set B; 3. f(cX(A)) ⊆ cY (f(A)) for any A ⊆ X; and 4. cX ( f−1(B) ) ⊆ f−1(cY (B)) for any B ⊆ Y . C. Balingit, J. Benitez / Eur. J. Pure Appl. Math, 12 (4) (2019), 1553-1566 1556 Remark 2.6. If f : X → Y is a G -continuous map such that f(x) ∈ ⋃ G∈V G, then (X,GX) is strong. This is so since for each x ∈ X there is a GY -open set V containing f(x) and consequently a corresponding Ux containing x such that f(Ux) ⊆ V . This indicates that X ⊆ ⋃ x∈X Ux ⊆ X and (X,GX) is therefore strong. The notion of a (µ, ν)(j,k)-continuous map, where j, k = 1, 2 are distinct, analogously defined for a bigeneralized topological space seen in [6], may be extended to maps between an m-GT space and n-GT space: Definition 2.7. For a pair of distinct j, k where 1 ≤ j ≤ m and 1 ≤ k ≤ n, a function f : X → Y is (µ, ν)(j,k)-continuous at a point x if for each νk-open set V containing f(x) there is a µj-open set U containing x such that f(U) ⊆ V . If f is (µ, ν)(j,k)-continuous at all points x ∈ X, then we say f is (µ, ν)(j,k)-continuous. Example 2.8. Let m,n ∈ N where m ≥ n and consider the m-GT space (X,GX) where X and GX be the same as defined in Example 2.2. In this case, let Y = [0, n] and for each k = 1, . . . , n, define νk = {∅} ∪ {[0, n− t+ 1]; t = 1, . . . , k}. Observe that (Y,GY ) is an n-GT space. Define the map f : Y → X as f(y) = (n−y n , n ) . Let y ∈ Y and V be a µ1-open set containing f(y) = (n−y n , n ) . Then V is of the form R1 b , where b ≤ n. Now, Y = [0, n] is a νm-open set containing y with f([0, n]) = {(x, n) : 0 ≤ x ≤ 1} ⊆ R1 n ⊆ R1 b for all b ≤ n. Hence, f is (ν, µ)(m,1)-continuous at y and since y is arbitrary, f is in fact (ν, µ)(m,1)-continuous. In the next results, some relationships between G -continuity and (µ, ν)(j,k)-continuity are established: Theorem 2.9. A function f : X → Y is G -continuous at a point x ∈ X if and only if for each k = 1, . . . , n, there exists 1 ≤ j ≤ m such that f is (µ, ν)(j,k)-continuous at x. Proof. Suppose that f is G -continuous at x ∈ X, k ∈ {1, . . . , n} and Vk is a νk-open set containing f(x). Then Vk is GY -open and since f is G -continuous, we can find a GX -open set U containing x such that f(U) ⊆ Vk. Since U is GX -open, there is a 1 ≤ j ≤ m such that for some Hj ∈ µj , we have x ∈ Hj ⊆ U and f(Hj) ⊆ f(U) ⊆ Vk. Thus, f is (µ, ν)(j,k)-continuous at x ∈ X. Conversely, suppose that V is a GY -open set containing f(x). Then for some k, there is a Gk ∈ νk such that f(x) ∈ Gk ⊆ V . By assumption, for some 1 ≤ j ≤ m, we can find a µj-open (and hence, G -open) set U containing x such that f(U) ⊆ Gk ⊆ V . Thus, f is G -continuous at x. � Theorem 2.9 implies that to inspect for G -continuity at a point of the domain, we may simply find for each component GT of (Y,GY ) a corresponding component GT of (X,GX) for which the continuity in the sense of Definition 2.7 holds. Some properties of (µ, ν)(j,k)-continuity can be drawn by a simple extension of those that are enumerated in [6]: C. Balingit, J. Benitez / Eur. J. Pure Appl. Math, 12 (4) (2019), 1553-1566 1557 Theorem 2.10. Let f : X → Y be a map. The following statements are equivalent: 1. f is (µ, ν)(j,k)-continuous at a point x ∈ X; 2. x ∈ iµj (f−1(V )) for each νk-open set V containing f(x); 3. x ∈ iµj (f−1(B)) for each B ⊆ Y such that x ∈ f−1(iνk(B)); and 4. x ∈ f−1(F ) for all νk-closed set F such that x ∈ cµj (f−1(F )). Theorem 2.11. Let f : X → Y be a map. The following statements are equivalent: 1. f is (µ, ν)(j,k)-continuous; 2. f−1(V ) = iµj (f −1(V )) for each νk-open set V ; 3. f−1(iνk(B)) ⊆ iµj (f−1(B)) for each B ⊆ Y ; and 4. cµj (f −1(F )) = f−1(F ) for all νk-closed set F . In view of Definition 2.7, (µ, ν)(j,k)-continuity may occur for each pair of indices j and k; and thence we have another type of continuity formally stated as follows: Definition 2.12. Let (X,GX) and (Y,GY ) be m-GT and n-GT spaces, respectively, where GX = {µ1, . . . , µm} and GY = {ν1, . . . , νn} for some m,n ∈ N. A function f : X → Y is called pairwise (µ, ν)-continuous if for each pair j, k where 1 ≤ j ≤ m and 1 ≤ k ≤ n, f is (µ, ν)(j,k)-continuous. Example 2.13. 1. Let (X,G1) and (X,G2) be strong m-GT and n-GT spaces, respec- tively, over the same set X and c ∈ X. If f : X → X is the constant function defined by f(x) = c, then for each k = 1, . . . , n and for each νk-open set Vk containing c, f(Gj) ⊆ {c} ⊆ Vk for all sets Gj ∈ µj . Thus, f is (µ, ν)(j,k)-continuous at any point x ∈ X and any pair j, k. As a result, f is pairwise (µ, ν)-continuous. 2. In Example 2.8, we observe that for a pair s, t ∈ N where 1 ≤ s ≤ n and 1 ≤ t ≤ m and for y ∈ Y , 0 ≤ n−y n ≤ 1 which means that f(y) ∈ R1 0 and so if t 6= 1, f vacuously satisfies (ν, µ)(s,t)-continuity on y. Now, if t = 1, we note that Y = [0, n] is νs-open for all s = 1, . . . , n and recall that the µ1-open set containing f(y) are precisely the sets R1 b where b ≤ n with f([0, n]) ⊆ R1 b . This indicates that f is (ν, µ)(s,t)- continuous at y. From these cases, we can see that with the arbitrary nature of y, f is pairwise (ν, µ)-continuous. 3. Consider the graph D∗ and the n-GT spaces (V (D∗),GV ) and (E(D∗),GE) as de- scribed in Example 2.3. Define the mapping g : E(D∗) → V (D∗) by g(e) = u for each e = uv ∈ E(D∗). Let j, k ∈ {1, . . . , n} and e ∈ E(D∗). If e = ei for some i = 1, . . . , s − 1, then g(e) = g(ei) = vi. Notice that µk-open sets V containing vi must contain either both vi−1 and vi or both vi and vi+1. For the former case, we take the νj-open set C. Balingit, J. Benitez / Eur. J. Pure Appl. Math, 12 (4) (2019), 1553-1566 1558 Figure 1: Directed graph D∗ for Example 2.13 (3). U1 = {ei−1, ei}, and for the latter, take U2 = {ei}. These sets both contain e = ei and since g(U1) = {vi−1, vi} ⊆ V and g(U2) = {vi} ⊆ V . On the other hand, if e = e∗i for some i = 1, . . . , n, then g(e) = vs for each µk-open set V containing vs, U = {ej} is νj-open and g(ej) = {vs} ⊆ V . Hence, g is (ν, µ)(j,k)-continuous at e ∈ E(D∗). With j, k and e arbitrary, we see that g is pairwise (ν, µ)-continuous. If every µj is a strong GT on X, and f : X → Y is defined by f(x) = c for some c ∈ Y , then for a pair of fixed indices j, k, and for a νk-open set Vk containing c, f(Uj) = {c} ⊆ Vk for all µj-open set Uj . Since each µj is strong, there is such µj-open set U∗j containing x whose image is contained in Vk. With j, k held arbitrary, we see that f is (µ, ν)-continuous. If, on the other hand, f : X → Y is a map such that f(X) ⊆ Y \ ( ⋃ G∈V G ) 6= ∅, then f is immediately pairwise (µ, ν)-continuous. Remark 2.14. If f : X → Y is pairwise (µ, ν)-continuous, x ∈ X and V is a G -open set containing f(x), then for some k there exists a νk-open subset Gk of V such that f(x) ∈ Gk. For any j, there is a µj-open (hence, GX-open) set U such that x ∈ U and f(U) ⊆ Gk ⊆ V . Thus, f is G -continuous whenever it is pairwise (µ, ν)-continuous. The next theorem provides some characterizations for a pairwise (µ, ν)-continuous map: Theorem 2.15. Let f : X → Y be a map. The following statements are equivalent: 1. f is pairwise (µ, ν)-continuous; 2. f−1(V ) is µj-open for each GY -open set V and for each j = 1, . . . ,m; 3. f−1(iGY (B)) ⊆ iµj (f−1(B)) for all B ⊆ Y and for all j = 1, . . . ,m; and 4. f−1(F ) is µj-closed for all GY -closed sets F . Proof. (1) ⇒ (2) If f is pairwise (µ, ν)-continuous and V is a GY -open set, then V = iGY (V ) = n⋃ k=1 iνk(V ). Now, each iνk(V ) ∈ νk. By Theorem 2.11 (2), for each j = 1, . . . ,m, f−1(iνk(V )) = iµj (f −1(iνk(V ))) and is therefore µj-open. As a result, f−1(V ) = C. Balingit, J. Benitez / Eur. J. Pure Appl. Math, 12 (4) (2019), 1553-1566 1559 n⋃ k=1 f−1(iνk(V )) is µj-open for every j = 1, . . . ,m. (2) ⇒ (3) Suppose that B ⊆ Y . Then iGY (B) is GY -open. By assumption, f−1(iGY (B)) is µj-open for each j = 1, . . . ,m, implying that f−1(iGY (B)) = iµj (f −1(iGY (B))) ⊆ iµj (f −1(B)) for each j = 1, . . . ,m. (3) ⇒ (1) For a pair j, k, and for B ⊆ Y , iνk(B) is νk-open and is therefore GY -open. Thus, by statement (3), f−1(iνk(B)) = f−1(iGY (iνk(B))) ⊆ iµj (f−1(iνk(B))) ⊆ iµj (f−1(B)). By the equivalence in Theorem 2.11, and since j, k are arbitrarily chosen, we see that f is pairwise (µ, ν)-continuous. (4) ⇔ (2) This equivalence follows from the fact that f−1(Y \F ) = X \f−1(F ) for any F ⊆ Y . � 3. G -open and G -closed maps Now, we define and examine G -open and G -closed maps on n-GT spaces: Definition 3.1. Let (X,GX) and (Y,GY ) be m-GT and n-GT spaces, respectively, where GX = {µ1, . . . , µm} and GY = {ν1, . . . , νn} for some m,n ∈ N and let f : X → Y be a map. f is called G -open map [resp. G -closed map] if f(A) is GY -open [resp. GY -closed] for each GX-open [resp. GX-closed] set A. Example 3.2. Consider the following examples: 1. Let (X,GX) and (Y,GY ) and f : X → Y be as defined in Example 2.2. If (a, b) ∈ X, then there is a j0 such that j0 − 1 < a ≤ j0. Define f∗ : X → Y such that f∗(a, b) = j0−a j0 . For each j = 1, . . . ,m and for each s ≥ 0, f∗(Rjs) = [0, 1], which is a GY -open set. But each GX -open set U ⊆ X is the union of sets Rjs; thus, f∗ is a G -open map. However, for the GX -closed set F = {(x, y) : 0 ≤ x ≤ 1, 0 ≤ y < n}, we see that f∗(F ) = [0, 1] is not a GY -closed set. Hence, f∗ is not a G -closed map. 2. Recall the spaces (V (D),GV ) and (E(D),GE) described in Example 2.3. Observe that each u ∈ V (D) is in some maximal path Pi in D. Let s = max{i : u ∈ V (Pi)} and denote Ps = [u(s,1), . . . , u(s,k)]. Then for every u ∈ U , there corresponds a unique pair (s, d) such that u = u(s,d) for some 1 ≤ d ≤ k. Define the map ϕ∗ : V (D) → E(D) by ϕ∗(u) = { e(s,k−1) = u(s,k−1)u(s,k), if d = k e(s,d) = u(s,d)u(s,d+1), if d 6= k . By the uniqueness of s, we see that ϕ∗ is a well-defined map. Also, by definition of the n-GT space (E(D),GE), {e} is GE-open for every arc e ∈ E(D). Hence, if U ⊆ V (D) is GV -open, then ϕ∗(U) = ⋃ u∈U ϕ∗(u) is GE-open. In turn, ϕ∗ : V (D) → E(D) is a C. Balingit, J. Benitez / Eur. J. Pure Appl. Math, 12 (4) (2019), 1553-1566 1560 G -open map. Meanwhile, for any GV -closed set F , we see that ϕ∗(F ) is definitely a GE-closed set. That is, ϕ∗ is a G -closed map. G -open and G -closed maps naturally satisfy the following inherent properties of open and closed maps in the sense of the ordinary topological spaces: Theorem 3.3. Let f : X → Y be a map and S ⊆ Y . 1. If f is a G -open map and A is a GX-closed set containing f−1(S), then there is a GY -closed set B containing S such that f−1(B) ⊆ A. 2. If f is a G -closed map and U is a GX-open set containing f−1(S), then there is a GY -open set set V containing S such that f−1(V ) ⊆ U . Corollary 3.4. If f : X → Y is a G -closed map, y ∈ Y and U is a GX-open set such that f−1({y}) ⊆ U , then y ∈ iY (f(cX(U))). Theorem 3.5. Let f : X → Y be a map. The following statements are equivalent: 1. f is a G -open map; 2. f(iX(A)) ⊆ iY (f(A)) for every A ⊆ X; and 3. For each x ∈ X and for every GX-open set U containing x, there exists a GY -open set W containing f(x) such that W ⊆ f(U). Theorem 3.6. A map f : X → Y is a G -closed map if and only if cY (f(A)) ⊆ f(cX(A)) for each A ⊆ X. Theorem 3.7. Let f : X → Y be a map. The following statements are equivalent: 1. f is a G -closed map; 2. If A ⊆ X is a GX-open set, then S = {y : f−1({y}) ⊆ A} is a GY -open set; and 3. If B ⊆ X is a GX-closed set, then T = {y : f−1({y}) ∩B 6= ∅} is a GY -closed set. We now establish the equivalence of G -open and G -closed maps involving bijective mappings: Theorem 3.8. If f : X → Y is bijective, then f is a G -open map if and only if f is a G -closed map. Proof. Observe that for a subset U of X, f(X\U) = Y \f(U) if and only if f is bijective. Thus, if f is a G -open map and U is G -closed, then f(X\U) = Y \f(U) is G -open so that f(U) is G -closed. That is, f is also a G -closed map. Similarly, if f is a G -closed map, then f is also a G -open map. � Definition 3.9. Let (X,GX) and (Y,GY ) be m-GT and n-GT spaces, respectively, where GX = {µ1, . . . , µm} and GY = {ν1, . . . , νn} for some m,n ∈ N. Furthermore, let f : X → Y be a map. C. Balingit, J. Benitez / Eur. J. Pure Appl. Math, 12 (4) (2019), 1553-1566 1561 1. f is called a (µ, ν)(j,k)-open map [resp. (µ, ν)(j,k)-closed map] if f(U) is a νk-open [resp. νk-closed] set for every µj-open [resp. µj-closed] set U ⊆ X. 2. f is called a pairwise (µ, ν)-open map [resp. pairwise (µ, ν)-closed map] if for each pair j, k where 1 ≤ j ≤ m and 1 ≤ k ≤ n, f is a (µ, ν)(j,k)-open map [resp. (µ, ν)(j,k)-closed map]. Notice that if both (X,GX) and (Y,GY ) are 1-GT spaces, then it immediately follows that f : X → Y is a G -open map if and only if f is a (µ, ν)(1,1)-open map. Furthermore, a (µ, ν)(j,k)-open map [resp. (µ, ν)(j,k)-closed map] may not be a G -open map [resp. G - closed map]. On the other hand, a G -open map [resp. G -closed map] may also not be a (µ, ν)(j,k)-open map [resp. (µ, ν)(j,k)-closed map]. These are illustrated in the following example: Example 3.10. Let X and Y be infinite sets and {P1, . . . , Pm} and {Q1, . . . , Qn} be some partitions of X and Y , respectively. Putting GX = {µj : j = 1, . . . ,m} and GY = {νk : k = 1, . . . , n} where µj = P(Pj) and νk = P(Qk), we come up with the m-GT space (X,GX) and the n-GT space (Y,GY ). 1. If, in general, f : X → Y is the constant map f(x) = c, then for the fixed k∗ (where c ∈ Qk∗), f is a (µ, ν)(j,k ∗)-open map and, in fact, also a G -open map. 2. In particular, consider the case where X = Y and m = n, while Pj 6= Qk for any j, k. Then if f : X → Y is the identity map, it is easy to see that f is a G -open map. However, f is not a (µ, ν)(j,k)-open map for any pair j, k due to our choice of partitions. 3. Now, consider the case where X 6= Y , ∅ 6= A ⊂ X and ∅ 6= B ⊂ Y , µ1 = P(A), µ2 = P(X\A) and ν1 = P(B). If GX = {µ1, µ2} and GY = {ν1}, then (X,GX) is a 2-GT space and (Y,GY ) is a 1-GT space. Furthermore, if c1 ∈ B and c2 ∈ Y \B are fixed and f : X → Y is defined as f(x) = { c1, if x ∈ A c2, if x ∈ X\A , then f is a (µ, ν)(1,1)-open map but not a G -open map since f(X) = {c1, c2} is not GY -open. Example 3.11. To illustrate a pairwise (µ, ν)-open map, we simply recall the mapping f in Example 3.2 (1.ii). This is so since the image of each G -open set (and hence of any µj-open set) is [0, 1] which is νk-open for all k. A (µ, ν)(j,k)-open map and a (µ, ν)(j,k)-closed map inherits similar and corresponding properties as established in the previous results in this section since a (µ, ν)(j,k)-open map and a (µ, ν)(j,k)-closed maps are analogous to a G -open map and a G -closed map, respec- tively, when the underlying spaces involved are both 1-GT spaces. C. Balingit, J. Benitez / Eur. J. Pure Appl. Math, 12 (4) (2019), 1553-1566 1562 We now present some observations showing relationships of the mentioned typed of open maps: Theorem 3.12. If for each j there is a k such that f is a (µ, ν)(j,k)-open map, then f is a G -open map. Proof. Let U be a GX -open set. Then for each x ∈ U , there exists a j and a µj-open set Ox such that x ∈ Ox ⊆ U . By assumption, there exists a k such that f(Ox) is νk-open. As a result, f(U) = f( ⋃ x∈U Ox) = ⋃ x∈U f(Ox) is a GY -open set. Since U is arbitrary, we say that f is a G -open map. � The converse of Theorem 3.12 is not generally true as seen in Example 3.10 (2). Corollary 3.13. If f is a pairwise (µ, ν)-open map, then f is a G -open map. Proof. This is immediate from the definition of a pairwise (µ, ν)-open map and then ap- plying Theorem 3.12. � Theorem 3.14. A mapping f : X → Y is a pairwise (µ, ν)-open map if and only if for each j and for each µj-open set U , f(U) ∈ n⋂ k=1 νk. Proof. Let f : X → Y be a map, j ∈ {1, . . . ,m} and U be a µj-open set. If f is a pairwise (µ, ν)-open map, then for every k, f(U) is µk-open so that f(U) ∈ n⋂ k=1 νk. The converse is similarly outright. � Corollary 3.15. If for each j and for each µj-open set U , f(U) ∈ n⋂ k=1 νk, then f is a G -open map. Proof. This relationship is a direct consequence of Theorem 3.14 and Corollary 3.13. � 4. G -homeomorphisms This section basically displays the relationships of G -continuous maps to G -open and G -closed maps in form of G -homeomorphisms. In the sense of ordinary topological spaces, the existence of homeomorphisms between two topological spaces identifies an equivalence of their structures. Here, we investigate whether the existence of such likeness extends this time to n-GT spaces. Definition 4.1. Let (X,GX) and (Y,GY ) be m-GT and n-GT spaces, respectively, where GX = {µ1, . . . , µm} and GY = {ν1, . . . , νn} for some m,n ∈ N. C. Balingit, J. Benitez / Eur. J. Pure Appl. Math, 12 (4) (2019), 1553-1566 1563 1. A bijective map f : X → Y is a G -homeomorphism if both f and f−1 are G - continuous maps. In this case, we use the notation f : X G∼= Y to denote that f is a G -homeomorphism. 2. If there exists a G -homeomorphism f : X G∼= Y , then we say that the spaces X and Y are G -homeomorphic. Example 4.2. By the definition of the spaces (X,G1) and (X,G2) in Example 3.10 (2), we see that the identity map between these particular spaces easily provides a G - homeomorphic map. In a wider sense, if f : X → Y is any bijection and (X,GX) and (Y,GY ) are as defined in Example 3.10, then f is in fact a G -homeomorphism. However, it should be noted that not every bijective mapping is a G -homeomorphism: see this by simply dropping Qn (or any of the component GTs) in the n-GT space (Y,GY ) in the same example. Example 4.2 suggests that even if f : X G∼= Y , it is not a guarantee that the spaces X and Y are equivalent in terms of their component GTs. In other words, a component GT µj in GX may not coincide to any νk in GY even if X is G -homeomorphic to Y . Theorem 4.3. Let f : X → Y be a bijective map. Then the following statements are equivalent: 1. f is a G -homeomorphism; 2. f is a G -continuous and G -open map; 3. f is a G -continuous and G -closed map; 4. f(cX(A)) = cY (f(A)) for every A ⊆ X; and 5. f(iX(A)) = iY (f(A)) for every A ⊆ X. Proof. Suppose f : X → Y is a bijective map. (1) ⇔ (2) Note that f is a G -homeomorphism if and only if f is G -continuous and f−1 is also G -continuous. As such, Theorem 2.4 provides that for every GX -open set U , (f−1)−1(U) = f(U) is GY -open. This means that f is also G -open map, and con- versely. (2)⇔ (3) Because f is bijective, these directions follow immediately from Theorem 3.8. (3)⇔ (4) This follows immediately from Theorems 2.5 and 3.6. (2) ⇔ (5) From Theorem 3.5, f is a G -open map if and only if f(iX(A)) ⊆ iY (f(A)). Also, since f is both bijective and G -continuous, f−1(iY (f(A))) is GX -open and f−1(iY (f(A))) ⊆ f−1(f(A)) = A implying that f−1(iY (f(A))) ⊆ iX(A). As a result, iY (f(A)) ⊆ f(iX(A)). Conversely, we only need to show that f−1(B) is GX -open for any GY -open set B whenever (5) holds. Indeed, if B is GY -open, (5) provides that f(iX(f−1(B))) = iY (f(f−1(B))) = iY (B) = B. C. Balingit, J. Benitez / Eur. J. Pure Appl. Math, 12 (4) (2019), 1553-1566 1564 Thus, iX(f−1(B)) = f−1(B) is a GX -open set which, in turn, shows that f is also G - continuous. � Definition 4.4. Let (X,GX) and (Y,GY ) be m-GT and n-GT spaces, respectively, where GX = {µ1, . . . , µm} and GY = {ν1, . . . , νn} for some m,n ∈ N and let f : X → Y be a bijective map. 1. f is called a (µ, ν)(j,k)-homeomorphism if f is a (µ, ν)(j,k)-continuous map and f−1 is a (ν, µ)(k,j)-continuous map and we write f : X (j,k)∼= Y . 2. f is called a pairwise (µ, ν)-homeomorphism if for each pair j, k, f is a (µ, ν)(j,k)- homeomorphism. In this case, we write f : X pw∼= Y . By the definition of (µ, ν)(j,k)-homeomorphism, it is easy to see that f is pairwise (µ, ν)-homeomorphism if and only if f is a pairwise (µ, ν)-continuous map and f−1 is a pairwise (ν, µ)-continuous map. Theorem 4.5. If f is a pairwise (µ, ν)-homeomorphism, then f is a G -homeomorphism. Proof. If f is a pairwise (µ, ν)-homeomorphism, then f is a pairwise (µ, ν)-continuous map and f−1 is a pairwise (ν, µ)-continuous map. By Remark 2.14, f and f−1 are G -continuous maps. Thus, f is a G -homeomorphism. � Theorem 4.6. If f is a pairwise (µ, ν)-homeomorphism, then µ1 = · · · = µm and ν1 = · · · = νn. Proof. We first show that if f : X (j,k)∼= Y for each pair j, k of indices, then {f(U) : U ∈ µj} = νk. Indeed, if U ∈ µj , then f(U) ∈ νk since f is also a (µ, ν)(j,k)-open as implied by Definition 4.4 and the equivalence of (1) and (2) in Theorems 2.11 and 2.15. Also, if V ∈ νk, f −1(V ) ∈ µj since f is (µ, ν)(j,k)-continuous, implying that V = f(f−1(V )) ∈ {f(U) : U ∈ µj}. Similarly, {f−1(V ) : V ∈ νk} = µj . Now, if f : X pw∼= Y , then for a fixed µj , {f(U) : U ∈ µj} = νk for all k = 1, . . . , n. That is, ν1 = . . . = νn. 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