IHJPAS. 36 (3) 2023 

398 

This work is licensed under a Creative Commons Attribution 4.0 International License 

 

   

 

 

 

 

 

 

 
*Corresponding Author: stabraq.abd2103m@ihcoedu.uobaghdad.edu.iq 

 

 

 

 

Abstract 

 The structure of this paper includes introduction the definition of the nano topological space, 

which was defined by M. L. Thivagar, who defined the lower approximation of G and the upper 

approximation of G, as well as defined the boundary region of G and some other important 

definitions that were mentioned in this paper with giving some theories on this subject. Some 

examples of defining nano perfect mappings are presented along with some basic theories. Also, 

some basic definitions were presented that form the focus of this paper, including the definition of 

nano  pseudometrizable space, the definition of nano compactly generated space, and the definition 

of completely nano para-compact. In this paper, we presented images of nano perfect mappings 

with some definitions and important evidence related to them, then we presented inverse images 

of nano perfect mappings with related theories. 

 

Keywords: Nano topological space,nano continuous mappings,nano compact space,nano perfect 

mappings. 

 

1. Introduction 

         [1] introduced the concept of nano topological spaces with respect to a subset G of a universe 

U, which known as terms of approximations and boundary region of a subset of an universe using 

an equivalence relation on it and also known as nano-closed-sets, nano-interior and nano-closure. 

[2] introduced fibrewise IJ-Perfect bitopological spaces  and [3] introduced weak and strong forms 

of ω- perfect mappings. [4] introduced R𝛼-compactness on bitopological spaces. [5] introduced 

on cohomology groups of four-dimensional nilpotent associative algebras. [6,7,8,9,10] introduce 

some basic concepts that helped us build this work. [11,12 ,13,14]  introduce some types of 

mappings in bitopological spaces and Soft Simply Compact Spaces and other topics related to the 

doi.org/10.30526/36.3.3051 

Article history: Received 30 September 2022, Accepted 20 February 2022, Published in  July 2023. 

 

 

 

Ibn Al-Haitham Journal for Pure and Applied Sciences 

Journal homepage: jih.uobaghdad.edu.iq 

 

Some Results on Nano Perfect Mappings 

 E. A. Mohsin 

Department of Mathematics, College of Education for 

Pure Science Ibn Al-Haytham, University of 

Baghdad, Baghdad, Iraq. 

 

ihcoedu.uobaghdad.edu.iq@istabraq.abd2103m 

    Y. Y. Y𝐨us𝐢f  
Department of Mathematics, College of Education 

for Pure Science Ibn Al-Haytham, University of 

Baghdad, Baghdad, Iraq. 

 

yousif.y.y@ihcoedu.uobaghdad.edu.iq 

 
               M. El Sayed  

Najran University, College of Science and Arts, 

Department of Mathematics, 66445, Saudi Arabia. 

 

mebadris@nu.edu.sa 

 

https://creativecommons.org/licenses/by/4.0/
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mailto:istabraq.abd2103m@ihcoedu.uobaghdad.edu.iq
mailto:istabraq.abd2103m@ihcoedu.uobaghdad.edu.iq
http://en.uobaghdad.edu.iq/?page_id=15060
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mailto:yousif.y.y@ihcoedu.uobaghdad.edu.iq
mailto:mebadris@nu.edu.sa
mailto:mebadris@nu.edu.sa


IHJPAS. 36 (3) 2023 

399 
 

mentioned sources. In this paper, we introduce a nano perfect mappings and several related 

theorems.  

Definition 1.1 : [15] Let U be a non empty finite set of objects is said to be the universe and R be 

an equivalence relation on U is called indiscernibility relation. Then U is divided into disjoint 

equivalence classes. Components belonging to the same equivalence class are called the 

indiscernible with one another. The spouse (U, R) is called the approximation-space. Let G ⊆ U. 

Then, 

a) The lower approximation of G with reference to R is the set of all objects which can be 

categorized as G with reference to R and is symboly by LoR(G). That is, LoR(G) = 

⋃{R(G): R(G) ⊆ G, g ∈ U} where R(G) indicate the equivalence class determined by g ∈ 

U. 

b) The upper approximation of G with reference to R is the set of all objects which can be 

possibly classified as G with reference to R and is symboly by UpR(G). That is, UpR(G) 

= ⋃{R(G): R(G) ⋂ G ≠ ∅, g ∈ U}. 

c) The boundary region of G with reference to R is the set of all objects which can be 

classified neither as G nor as not /G with reference to R and is symboly by BoR(G). BoR(G) 

= UpR(G) / LoR(G). 

 

 Property 1.2 : [16] If (U, R) is an approximation space and G,H ⊆ U, then  

a) LoR(G) ⊆ G ⊆ UpR(G).  

b) LoR(∅) = UpR(∅) = ∅. 

c) LoR(U) = UpR(U) = U.  

d) UpR(G⋃H) = UpR(G) ⋃ UpR(H).  

e) UpR(G⋂ H) ⊆ UpR(G) ⋂ UpR(H) . 

f) LoR(G⋃H) ⊇ LoR(G) ⋃ LoR(H) . 

g) LoR(G⋂H) = LoR(G) ⋂ LoR(H). 

h) LoR(G) ⊆ LoR(H)  and  UpR(G) ⊆ UpR(H)  whenever  G⊆H. 

i) UpR(G
c) =[LoR(G)]

𝐶  and  LoR(G
c) = [UpR(G)]

c. 

j) UpR[UpR(G)] = LoR[UpR(G)] = UpR(G). 

k) LoR[LoR(G)] = UpR[LoR(G)] = LoR(G). 

   

 Definition 1.3 : [16] If τR(G) is the nano-top-sp.on U with respect to G, then the set B = { U , 

LoR(G)    , BoR(G) } is the nano-basis for τR(G). 

 

 Definition 1.4 : [17] Let U be the universe, R be an equivalence relation on U and τR(G) = {U, 

∅,  LoR(G), UpR(G), BoR(G)}, where G ⊆ U. Then τR(G) it achieves the following axioms:  

a) U and ∅ ∈ τR(G).  

b) The union of the components of any subset of τR(G) is in τR(G).  

c) The intersection of the components of any finite subset of τR(G) is in τR(G). 

Therefore, τR(G) is a topology on U called the nano topology (denoted by nano-top.) on U with 

reference to G. We call (U, τR(G)) as the nano topological space (denoted by nano-top-sp.) The 

components of τR(G) are said to be nano open sets (denoted by nano-ope-sets). The complement 

of a nano-ope-set is called a nano closed set(denoted by nano-clos-set). 



IHJPAS. 36 (3) 2023 

400 
 

Definition 1.5 : [18] Let (U, τR(G)) and (V, τR` (H)) be nano-top-sp. Then a mapping (dented by 

map.) f : (U, τR(G)) ⟶ (V, τR` (H)) is nano continuous (denoted by nano-cont.) on U if the inverse 

image of each nano-ope-set in V is nano-ope. in U.  

Definition 1.6 : [18] A function f : (U, τR(G)) ⟶ (V, τR` (H)) is a nano-clos. mapping if the image 

of each nano-clos-set in U is nano-clos. in V. 

Definition 1.7 : [13] Let (G, τR(G)) be a nano-top-sp.and A ⊆ G.The nano-closure of a set A is 

symboly by NCI(A) is the intersection of all nano-clos-sets containing A or all clos-super-sets of 

A; i.e.,the smallest clos-set containing A. 

Definition 1.8 : [18] A function f:(U,τR(G))⟶(V,τR` (H)) is called a nano homeomorphism if : 

a) f is 1-1 and onto. 

b) f is nano-cont. 

c) f is nano-ope. 

Definition 1.9 : [19] A set {Ai: i ∈ I } of nano-ope-sets in a nano-top-sp. ((U, τR(G)) is said to be 

nano-ope-cover of a subset B of U if B ⊂ {Ai: i ∈ I } holds.  

Definition 1.10 : [19] A space (U, τRʹ (G)) is called nano Hausdroff space (denoted by nano-Hausd-

sp.) if whenever g and h are distinct points of (U,τR(G)) , find disjoint nano-ope-sets A and B such 

as g ∈ A and h ∈ B. 

Definition 1.11 : [19] Let (U,τR(G)) be a nano-top-sp. and W be a subspace of G. We said to be 

space W is nano-comp-sp. iff each open-cover from G cover W has a finite-sub-cover. 

Definition 1.12 : [18] Let (G, τR(G)) be anano-top-sp., and let {Ui : Ui ⊂ G}i∈Λ be a open-cover 

of G is called locally finite-cover if for all point g ∈ G, find a nbd Ug of  g ⊂  Ug such as it 

intersects only limited many components of the cover, hence such as Ug ⋂ Ui  ≠  ∅, for only a 

finite number of i ∈ Λ. 

Definition 1.13 : [20] A top-sp.(G, τ) and the open-cover {Ui: Ui ⊂ G}i∈I is called refinement if a 

set of open subsets {Vj: Vj ⊂ G}i∈I
 which is still an open-cover in itself and such as for each j ∈ J, 

find an i ∈ I with Vj ⊂ Ui. 

Definition 1.14 : [21] A nano-top-sp.(G, 𝜏𝑅 (𝐺)) is called nano regular space (denoted by nano-

regu-sp.) iff for all nano closed set (denoted by nano-clos-set) F ⊂ G, and all point g ∉ F find nano 

open sets(denoted by nano-ope-sets) U and V such as g ∈ U, F ⊂ V and U⋂V = ∅.  

Definition 1.15 : [21] A anao-top-sp.(U,τR(G)) is called nano-normal-space (denoted by nano-

norm-sp.) if for any pair of disjoint nano-clos-sets A and B of U find nano-ope-sets V and W of U 

such as A ⊆ V and B ⊆ W. 

Definition 1.16 : A nano-top-sp.(G, 𝜏𝑅 (𝐺)) is said to be nano paracompact space (denoted by 

nano-para-comp-sp.) if it satisfies for each nano-ope-cover has a locally finite-nano-open 

refinement.  

Definition 1.17 : A nano-top-sp.(G, 𝜏𝑅 (𝐺)) is said to be nano metrizable-space if there is a metric 

d : G×G ⟶ [0,∞], such as the nano topology induced by d is τ, and is nano homeomorphic to a 

metric-space.  

2. Images of Nano Perfect Mappings. 

We study the Images of nano perfect mappings and some theories related to the topic. 

Definition 2.1 : A map f : G ⟶ H is said to be nano perfect (dented by nano-perf.) if it is nano 

continuous (denoted by nano-cont.), nano closed (denoted by nano-clos.), and for each H ∈ 

h, 𝑓 −1(h) is nano compact (denoted by nano-comp.).  



IHJPAS. 36 (3) 2023 

401 
 

We show by the next examples: 

Example 2.2 : Let f : U ⟶ V be a map such as U= {a, b, c, d} with U R⁄  = {{a},{b} 

, {c, d}}. Let G = {𝑎, 𝑐}  ⊂ 𝑈. Then τR(G) ={U, ∅ ,{a},{a,c,d},{c,d}}. Let V= {x, y, z, w} with 

V Rʹ⁄  = {{x},{y}, {z,w}} and H= {x,z} ⊂ V . Then τRʹ (H) = {V, ∅ ,{x},{x,z,w},{z,w}}. Such as 

f(a) = x , f(b) = y , f(c) = z , f(d) = w . Then f is nano-perf-map. 

Example 2.3 : Let U ={a,b,c,d} with U R⁄  = {{b},{c},{a,d}}. Let G = {a,b} ⊂ U . Then τR(G) ={ 

U, ∅ ,{b},{a,b,d},{a,d}}. Let V = {1, 2, 3, 4} with V Rʹ⁄  = {{1},{2},{3},{4}} and H = {1, 3, 4} ⊂ 

V . Then τRʹ (H) = {V, ∅ ,{1, 3, 4,}}. Define f : U ⟶ V such as f(a) = 1 , f(b) = 2, f(c) = 3, f(4) = 

4 then f is not nano-perf-map.  

Theorem 2.4 : Let f : G ⟶ H be a nano-perf-map. of G onto H . If the weight of G is infinite, then 

w(H) ≤ w(G). 

Proof: Let 
V
∽

 be a nano-base for the nano-topology of G such as |
V
∽

| = w(G) and let 
W
∽

 consist of 

every sets of the form ⋃ BB∈B⏟ , where 
B
∽ 

 is a finite subset of 
V
∽

 . If W ∈
w
̴
 , let TW = H\f(G\W) and 

let T be the set of nano-ope-sets of H of the form TW for some W in 
w
̴
. So w(G) is infinite, |

w
̴
| = 

w(G) , so that the cardinality of the set 
T
∽

 of nano-ope-sets of H does not exceed w(G). Furthermore 

T
∽

 is a nano-base for the nano-topology of H. For let J be an nano-ope-set of H and let h be a point 

of J. Since 𝑓 −1 (h) ⊂  𝑓 −1(J) and 𝑓 −1(h) is nano-comp., find  W of 
W
∽

 such as 𝑓 −1(h) ⊂ W ⊂

 𝑓 −1(J). Then h ∈  TW  ⊂ J. It follows that w(H)  ≤ w(G).  

Theorem 2.5 : The class of nano-Ti-spaces (denoted by nano-Ti-sp.) is invariant under nano-perf-

maps for i=1,2,3,4. 

Proof: Let f : G ⟶ H be a nano-perf. onto map: 

(a) If G is a nano-T1-sp. then all point {g} is a nano-clos-set. So let h ∈ H, then find g ∈ G such as 

f(g) = h. But f is nano-clos. and {g} is nano-clos.; hence {h} is nano-clos. in H. 

(b) let G be a nano-T2-sp. Let h1, h2 be a pair of distinct points of H. The inverse images 𝑓
−1(h1) 

and 𝑓 −1(h2) are nano-comp.and disjoint subsets of G, so find nano-ope-sets U, V ⊂ G such as 

𝑓 −1(h1) ⊂ U, 𝑓
−1(h2) ⊂ V and U⋂V=∅. So f is nano-clos., the sets A=H\f(G\U) and B=H\f(G\V) 

are nano-ope. in H such as 𝑓 −1(h1) ⊂  𝑓
−1(A) ⊂ U and 𝑓 −1(h2) ⊂  𝑓

−1(B) ⊂ V, A and B are 

nano-nbds of h1 and h2 respectively. Moreover A⋂ B =[H\f(G\U)] ⋂ [H\f(G\V)] 

 =H\[f(G\U) ⋃ f(G\V)] 

 =H\f[G\(U⋂V)] 

 =H\f(G) 

 =∅ 

So H is a nano-T2-sp. 

(c) Let G be a nano-T3-sp. Let h ∈ H and F be a nano-clos-set in H such as h ∈ H so 𝑓
−1(h) is 

nano-comp-sub-set of G and 𝑓 −1(F) is a nano-clos-sub-set of G and 𝑓 −1(h) ⋂𝑓 −1(F) = ∅. So find 

U, V nano-ope-sub-sets of G such as 𝑓 −1(h) ⊂ U and 𝑓 −1(F) ⊂ V with U⋂V = ∅. So f is nano-

clos., find A, B nano-ope-sub-sets of H such as 𝑓 −1(h) ⊂  𝑓 −1(A) ⊂ U and 𝑓 −1(F) ⊂  𝑓 −1(B) ⊂ 

V. Clearly, h ∈ A, F ⊂ B and A⋂B = ∅ so H is a nano-T3-sp. 

(d) Let G be a nano-T4-sp. Let F1, F2 be distinct nano-clos. subsets of H. So 𝑓
−1(F1) and 𝑓

−1 (F2) 

are nano-clos-sub-sets of G, find U, V nano-ope-sub-sets of G such as𝑓 −1(F1) ⊂ U, 𝑓
−1(F2) ⊂ V. 

Since f is nano-clos., find nano-ope-sub-sets A, B of H such as𝑓 −1(F1) ⊂  𝑓
−1(A) ⊂ U and 𝑓 −1(F2) 

⊂  𝑓 −1(B) ⊂ V. Clearly, F1 ⊂ A, F2 ⊂ B and A, B are disjoint, hence H is a nano-T4-sp.  



IHJPAS. 36 (3) 2023 

402 
 

 

Theorem 2.6 : Nano-compactness is invariant under nano-perf-maps.  

Proof: Let f : G ⟶ H be a nano-perf-map from a nano-comp-sp. G onto a space H. Let 
U
∽

 = { Uα 

: α ∈  Λ } be any nano-ope-cover of H. then 𝑓 −1( 
U
∽

 ) is an nano-ope-cover of G. So find a finite-

sub-cover {𝑓 −1(Uαi )}i=1
n

 for G. Then {Uαi }i=1
n

 is a finite-sub-cover of H, hence H is nano-comp.  

Lemma 2.7 : For each nano-comp-sub-sp. A of a locally nano-comp-sp. G and each nano-ope-set 

V that contains A find an nano-ope-set U such as A ⊂ U ⊂ NCl(U) ⊂ V and NCl(U) is nano-comp. 

Proof: For each g ∈ A take a nano-nbd  Vg of the point g such as NCl(𝑉g)  ⊂ V and a nano-nbd Wg 

of g such as NCl(𝑊g) is nano-comp. The set NCl(Ug), where Ug= Vg⋂Wg is nano-comp. because 

it is a nano-clos-sub-set of the nano-comp-sp. NCl(Wg). So find a finite set {g1, g2, … , gk } ⊂ A 

such as A ⊂ U = Ug1  ⋃ Ug2  ⋃ … ⋃ Ugk . The set NCl(U) = NCl(Ug1 ) ⋃ NCl(𝑈𝑔2 ) ⋃ … 

⋃ NCl(Ugk ) is nano-comp. and clearly we have NCl(U)  ⊂  NCl(Vg1 ) ⋃ NCl(Vg2 ) ⋃ … 

⋃ NCl(Vgk )  ⊂ V.  

Theorem 2.8 : Locally nano-compactness is invariant under nano-perf-maps.  

Proof: Let f : G ⟶ H be a nano-perf-map of a locally nano-comp-sp. G onto a space H. By Lemma 

2.7, for each  h ∈ H find an nano-ope-set V ⊂ G such as 𝑓 −1(h) ⊂ V and NCl(V) is nano-comp. 

The set W = H\f(G\V) is a nbd of h and since W = H\f(G\V) ⊂ H\(G\NCl(V)) ⊂ f(NCl(V)), the 

nano-closure NCl(W) is a nano-comp-sub-sp.of H.  

Theorem 2.9 :  A nano-cont. map f : G ⟶ H is nano-clos. if and only if for each point h ∈ H and 

each nano-ope-set U ⊂ G such as 𝑓 −1(h) ⊂ U, then find an nano-ope-set V of H such as h ∈ V and 

𝑓 −1(V) ⊂ U. 

Proof : (⇒) Let f be a nano-cont. map f : G⟶ H is nano-clos. and g ∈ H. Let U be an nano-ope-

set in G, such as 𝑓 −1(h) ⊂ U, so G \ U is nano-clos. in G. This implies f(G \ U) is nano-clos. in H. 

Let V = H \ f(G \ U), then V is an nano-ope-sub-set of H such as h ∈ V and 𝑓 −1(V) = G \ 𝑓 −1(f(G 

\ U)) ⊂ U. 

 (⇐) Assume that the assumption holds. Let F be a nano-clos-sub-set of G and h ∈ H \ f(F) and 

each nano-ope-set U ⊂ G such as 𝑓 −1(h) ⊂ U find an nano-ope-set V of H such as h ∈ V and 

𝑓 −1(V) ⊂ U. It is easy to show that V ⊂ H \ f(F). Hence H \ f(F) is nano-ope. in H and this implies 

f(F) is nano-clos. in H. 

Lemma 2.10 : If f : G ⟶ H is a nano-perf. onto map and {Aα}α∈Λ is a locally finite family of 

subsets of G, then {𝑓(Aα)}α∈Λ is a locally finite family of subsets of H.  

Proof: Let h be a point of H. If g ∈  𝑓 −1(h) , then find an nano-ope-set Vg of G such as g ∈  Vg and 

Λg = { α ∈  Λ : Vg⋂Aα  ≠  ∅ } is finite. Since 𝑓
−1(h) is nano-comp., find a finite subset B of 

𝑓 −1(h) such as 𝑓 −1 (h) ⊂  ⋃g∈B Vg. By Theorem 2.9, find an nano-ope-set W of H such as h ∈ W 

and 𝑓 −1(W) ⊂  ⋃g∈BVg. If W⋂f(Aα) ≠ ∅, then 𝑓
−1(W) ⋂Aα ≠ ∅, so that α ∈ M= ⋃g∈B. Λg Since 

M is a finite subset of Λ, it follows that the family {𝑓(Aα)}𝛼∈Λ is locally finite. 

Theorem 2.11 : For each point-finite nano-ope-cover {Us}s∈S of a nano-normal space G find an 

nano-open-cover {Vs}s∈S of G such as NCl(Vs)  ⊂  Us for each s ∈ S. 

Theorem 2.12 : If each nano-ope-cover of a top-sp. G has a locally finite nano-clos. refinement, 

then each nano-ope-cover of G has also a locally finite nano-ope. refinement. 

Theorem 2.13 : Nano-para-compactness is invariant under nano-perf-maps.  



IHJPAS. 36 (3) 2023 

403 
 

Proof: Let f : G ⟶ H be a nano-perf-map from a nano-para-comp-sp. G onto a nano-top-sp. H. 

Let 
V
∽

 be any nano-ope-cover of the space H. So G is nano-para-comp., {𝑓 −1(V)}V∈V⏟ has a locally 

finite nano-ope. improvement {Us}s∈S. Therefore G is norm-sp., therefore by Theorem 2.11,find a 

nano-clos-cover 
F
∼
 = {Fs}s∈S of G such as Fs  ⊂  Us for each s ∈ S. Since 

F
∼

 is locally finite and 

nano-clos., it follows that {𝑓(Fs)}s∈S is a locally finite nano-clos.improvement of the cover 
V
∼
 . 

Hence by Theorem 2.12, 
V
∼
 has an nano-ope. locally finite improvement and hence H is nano-para-

comp. 

Definition 2.14 : [15] Let 
A
~
 = {As}s∈S be a cover of a set G; the star of a set M ⊂ G with reference 

to 
A
∽

 is the set St(M, A) = ⋃ {As : M⋂As ≠ ∅ }. The star of one ̴ point set {G} with reference to a 

cover 
A
 ~

 is said to be the star of the point g with reference to 
A
~
 and is symboly by St(g, 

A
~
 ). 

Definition 2.15 : A nano-pseudometrizable-space is a pair (G,d) where G is a set and d is a function 

d:G×G⟶[0,∞], is said to be nano-pseudometrizable, that satisfies the following properties for all 

g,h,i in G: 

a) d(g,h) = 0 if and only if g=h. 

b) d(g,h) = d(h,g). 

c) d(g,h) ≤ d(g,i) + d(i,h). 

Lemma 2.16 : A space G is nano-pseudometrizable if and only if find a sequence {
F
~n

}
n∈N

 of 

locally finite nano-clos-covering of G such as for all point g of G and all nano-ope-set U such as 

g ∈ U find an integer n such as St(g, 
F
~n

 ) ⊂ U. 

Theorem 2.17 : If f : G ⟶ H is a nano-perf. onto map. and G is nano-pseudometizable-space, 

then H is a nano-pseudometizable-space. 

Proof: Choose a nano-pseudometric d on G which induces the nano-topology of G. For all positive 

integer n , let 
E
~𝑛

 be a locally finite nano-clos. improvement of the covering {B1
2n⁄

(G)}
g∈G

 of G. 

Then by Lemma 2.10, 
F
~n

 = { f(E) : E ∈  
E
~n

} is a locally finite nano-clos-covering of H for all 

positive integer n. We shall complete the proof by showing that the sequence {
F
~n

}
n∈N

 satisfies the 

condition of Lemma 2.16. Let J be an nano-ope-set of H and let h be a point of J. Then 𝑓 −1 (h) is 

nano-comp. and 𝑓 −1(h) ⊂  𝑓 −1(J). The nano-cont. real value function ∅ know in 𝑓 −1(h), by putting 

∅(g) =d(g, G\𝑓 −1(J)) if g ∈  𝑓 −1(h), is bounded and attains its bounds. Since ∅(g) >0 if g ∈

 𝑓 −1(h), it follows that find a point go  ∈  𝑓
−1(h) such as ∅ (g) ≥  ∅(g0) > 0 if g ∈  𝑓

−1(h). Thus 

find a positive integer n such as d(g, G\f(J)) ≥  1 2⁄
𝑛
 if g ∈  𝑓 −1(h). suppose that F 

F
~n+1

 and h ∈ F. 

Then F = f(E), where E ∈  
E
∼n+1

 and E⋂𝑓 −1(h) ≠  ∅. Let g be a point of E⋂𝑓 −1(h) and suppose 

find g2 ∈ G such as  g2 ∈ E⋂(G\𝑓
−1(J)). Since there g1 ∈ G such that E ⊂ B1

2⁄
n+1(g1), it follows 

that d(g, g2) ≥ d(g, g1) + d(g1, g2) <  
1

2⁄
𝑛
.  

Which is a inconsistency. Hence E ⊂  𝑓 −1(J) so that F ⊂ J. Thus St(h, 
F
∼n

) ⊂ J.  

Corollary 2.18 : If f : G ⟶ H is a nano-perf. onto map and G is a nano-metrizable-space, then H 

is a nano-metrizable-space. 

 Defintion 2.19  : In a nano-top-sp. G is a nano-compactly-generated-sp. (denoted by nano k-

space) in which a subset is nano-clos. if it is intersection with any nano-comp-sub-set is clos.  



IHJPAS. 36 (3) 2023 

404 
 

Lemma 2.20 : A nano-Hausd-sp. G is a nano k-space if and only if for all A ⊂ G, the set A is 

nano-clos. in G on condition the intersection of A with any nano-comp-sub-sp. F of the space G is 

nano-clos. in F. 

Theorem 2.21 : If there exists a nano-perf-map f : G ⟶ H of a nano k-sp. G onto a top-sp. H, then 

H is a nano k-sp.  

Proof: Let A ⊂ H such as A⋂F is nano-clos. in F for each nano-comp-sub-set F of H. It suffices 

to show that A is nano-clos. in H. Let K be a nano-comp-sub-set of G, so f(K) is nano-comp. in H, 

hence A⋂f(K) is nano-clos. in f(K), so 𝑓 −1(A)⋂K is nano-clos. in K. But G is a nano k-sp., so 

𝑓 −1(A) is nano-clos. in G. Thus 𝑓𝑓 −1(A) = A is nano-clos. in H, hence H is a nano k-sp. 

 3.  Inverse Images of Nana Perfect Mappings. 

 We study the Inverse Images of nano perfect mappings and several related theorems.  

Theorem 3.1 : Nano-regularity is an inverse of nano-perf-maps. 

Proof: Let f : G ⟶ H be a nano-perf-map onto a nano-regu-sp. H. Take a point g ∈ G and a nano-

clos-set F ⊂ G such as g ∉ F. The set F⋂𝑓 −1(f(g)) is nano-comp. and does not contain g, since G 

is nano-Hausd-sp. Find disjoint nano-ope-sets U1, V1 ⊂ G such as g ∈  U1 and F⋂𝑓
−1(f(g)) ⊂  V1. 

The set f(F\V1) is nano-clos. in H and does not contain f(g), hence by nano-regularity of H, find 

disjoint nano-ope-sets U2 , V2  ⊂ H such as f(g) ∈  U2 and f(F\V1) ⊂  V2. The sets U = U1⋂𝑓
−1(U2) 

and V = V1⋂𝑓
−1(V2) are nano-ope. in G, disjoint, and contain the point g and the set F alternately. 

Remark 3.2 : The remaining axioms of separation are not inverse invariants of nano-perf-maps. 

Also complete nano-regularity is not inverse invariant of nano-perf-maps. 

See [8] Exercis 3.10.c (c) , problem 3.12.20 (e) , Example 5.1.40, and the book. 

Theorem 3.3 : If f : G ⟶ H is a nano-perf-map, then for each nano-comp-sub-sp. R ⊂ H the 

inverse image 𝑓 −1(R) is nano-comp. 

Proof : Let Ṵ = { Uα : α ∈ Λ } be a family of nano-ope-sets of G such as 𝑓
−1(R) ⊂ ∪α∈Λ Uα. If h 

∈R, then find a finite subset M(h) of Λ such as 𝑓 −1(h) ⊂ ∪α∈M(h)  Uα. Since f is a nano-clos.map, 

by Theorem 2.6 find an nano-ope-set Vh of H such as h ∈  Vh and 𝑓
−1(Vh )  ⊂ ∪α∈M(h)  Uα. Since 

R is nano-comp., find a finite subset B of R such as R ⊂ ∪h∈B Vh . Hence 𝑓
−1(R) ⊂ ∪h∈B  𝑓

−1(Vh) 

⊂ ∪h∈B ∪α∈M(h)  Uα. Thus if M = ∪h∈B M(h), then M is a finite subset of Λ and 𝑓
−1(R) ⊂

 ∪α∈M  Uα. Thus 𝑓
−1(R) is nano-comp. 

Theorem 3.4 : Nano-compactness and local nano-compactness are inverse invariants of nano-

perf-maps.  

Proof: The inverse invariance of nano-compactness follows directly from Theorem 3.3 . If f : G 

⟶ H is a nano-perf-map and H is a locally nano-comp-sp., then for each g ∈ G, find a nbd U ⊂ G 

such as f(U) is contained in a nano-comp-sub-sp. I of the space H. Since f(NCl(U)) ⊂

 NCl(f(U))  ⊂ R, the set NCl(U)  ⊂  𝑓 −1(R) is nano-comp. 

Theorem 3.5 : Nano-para-compactness is an inverse invariant of nano-perf-maps.  

Proof: Let f : G ⟶ H be a nano-perf-map. onto a nano-para-comp-sp.H. Consider an nano-ope-

cover {Us}s∈S of the nano-space G and for each h ∈ H choose a finite set S(h) ⊂ S such as 𝑓
−1(h) 

⊂  ⋃ Uss∈S(h) . Since f is nano-clos., by Theorem 2.9, find a nbd 𝑉ℎ of the point h such as 𝑓
−1(h) 

⊂  𝑓 −1(Vh) ⊂  ⋃ Uss∈S(h) . The nano-ope-cover {Vh}h∈H of H has an nano-ope. locally finite 

improvement {Wt}t∈T. The family {𝑓
−1(Wt)}t∈T is an nano-ope. locally finite-cover of G and for 

each t ∈ T find a ht  ∈ H satisfying 𝑓
−1(Wt) ⊂  𝑓

−1(Vht ) ⊂  ⋃ Uss∈S(ht) . So the family { 𝑓
−1(Wt) 

⋂ Us : t ∈ T and s ∈ S(ht) } is an nano-ope. locally finite improvement of {Us}s∈S. 



IHJPAS. 36 (3) 2023 

405 
 

Theorem 3.6 : The Cartesian product G×H of a nano-para-comp-sp. G and a nano-comp-sp. H is 

nano-para-comp. 

Proof: Consider the projection p : G×H ⟶ H. Since H is nano-comp., then p is nano-perf. Since 

H is nano-para-comp., therefore by Theorem 3.5, G×H is nano-para-comp.  

Definition 3.7 : [22] A family {Bα: α ∈ Γ } of subsets of a space G is said to be a weak 

improvement of covering 
U
∼
 of G if find a subset Γ` of Γ such as {Bα: α ∈  Γ

`} is a covering of G 

and a improvement of  
U
∼

 .  

Definition 3.8 : A nano-top-sp. G is called completely nano-para-comp. if all nano-ope-covering 

of G has a weak improvement of the form { Vα: α ∈  Λ } where Λ = ⋃n∈NΛn and { V𝛼 : α ∈  Λn } 

is a star finite nano-ope-covering of G for all n.  

Theorem 3.9 : Let f : G ⟶ H be a nano-perf. onto map. If H is a completely nano-para-comp-sp., 

then G is a completely nano-para-comp-sp.  

Proof: Let  
U
~
 = { Uα: α ∈  Λ } be an nano-ope-cover of G. If h ∈ H, find a finite subset Λ(h) of Λ 

such as 𝑓 −1(h) ⊂  ⋃α∈Λ(h)Uα . So f is a nano-clos. map, find an nano-ope-set Wh of H such as h 

∈  Wh and 𝑓
−1(Wh) ⊂  ⋃𝛼∈Λ(h)Uα, and 

W
∼

 = { Wh: h ∈ H } is an nano-ope-covering of H. So H is 

completely nano-para-comp., find a family a {Vβ: β ∈  Γ } of nano-ope-sets of H such that : 

(a) Γ = ⋃n∈N Γ(n) and the family { Vβ: β ∈  Γ(n) } is a star finite-covering of H for all  n, and  

(b) Find a subsets Γ` of Γ such as {Vβ: β ∈  Γ
`} is a improvement of 

W
∽

 .  

If β ∈  Γ`, choose h(β) in H such as Vβ ⊂ Wh(β). Let O be the family of nano-ope-sets of G which 

consists of each sets 𝐹−1(Vβ)⋂Uα, where β ∈  Γ
`and α ∈  Λ(h(β)), together with each sets 

𝑓 −1(Vβ), where β ∈  Γ\Γ
`. Then O consists of countably many star finite nano-ope-coverings of G 

and O is a weak improvement of 
U
∽

 since the subfamily { 𝑓 −1(Vβ)⋂Uα: α ∈ Λ(h(β)), β ∈ Γ
`} is a 

covering of G and a improvement of U. Then if H is completely nano-para-comp., then G is 

completely nano-para-comp.  

Definition 3.10 : Let G be a nano-Hausd-sp.and let C be a family consisting of those subsets of G 

that have nano-clos.interssections with all nano-comp-sub-sps.of G. The set G with the nano-

topology generated by the family C of nano-clos-subsets will be symboly by kG is nano-ope. if 

and only if its intersection with any nano-comp-sub-sp. R of the space G is nano-ope.in R. The 

nano-topology of kG is finer than the nano-topology of G. 

Proposition 3.11 : The nano-spaces kG and G have the same nano-comp-sub-sps.  

Proposition 3.12 : Define for each nano-cont. map f: G ⟶ H where G and H nano-Hausd-sps., 

the map kf assigning to g ∈ kG the point f(g) ∈ kH . And Ǥg: kG ⟶ G by Ǥg(g) = g. Then:  

(a) Ǥg: kG ⟶ G is nano-cont.  

(b) kf: kG ⟶ kH is nano-cont.  

(c) kf satisfies the equality foǤg = ǤHokf.  

Theorem 3.13 : For a nano-cont. map f : G ⟶ H of a nano-Hausd-sp. G to a nano k-sp. H the 

statement are equivalent :  

(a) The map. f is nano-perf.  

(b)  For each nano-comp-sub-sp. R ⊂ H the restriction 𝑓I : 𝑓
−1(R) ⟶ R is nano-perf.  

(c)  For each nano-comp-sub-sp. R ⊂ H the inverse image 𝑓 −1(R) is nano-comp.  

Proposition 3.14 : If the composition gof of nano-cont. map f : G ⟶ H and j : H ⟶ R, where H 

is a nano-Hausd-sp., is nano-perf., then the map j│f(g) and f are nano-perf . 



IHJPAS. 36 (3) 2023 

406 
 

Proof : (a) For each point r ∈ R the fiber  (j │f(G))
−1 (r) =f(G) ⋂ j−1(r) = f( (jof)−1(r)) is nano-

comp., because the fiber (jof )−1(r) is nano-comp. Any nano-clos. subset of f(G) is of the form 

A⋂f(G), where A is nano-clos. in H. As the inverse image 𝑓 −1(A) is nano-clos. in G and jof is a 

nano-clos. map, the set (j│f(G)) (A⋂f(G)) = j(A⋂f(G)) = jof (𝑓
−1(A)), is nano-clos. in R, we have 

j│f(G) is a nano-clos. map and thus the map j│f(G) is a nano-clos. map and thus the map j│f(G) is 

nano-perf. 

(b) For each point h ∈ H the fiber 𝑓 −1(h) = [ ( jof)−1(j(h)]⋂ 𝑓 −1(h) is nano-comp. To conclude 

the proof it suffices to show that f is nano-clos. For each nano-clos-set F ⊂ G the map (jof)│F is 

nano-perf., so that by the first section of us proof the restriction j│f(F) is nano-perf. ; since the latter 

map ability be continuously extended through NCl(f(F)), it follows by Lemma 2.11 that f(F) = 

NCl(f(F)), and thus f is a nano-clos. map. 

Theorem 3.15 : If find a nano-perf- map f : G ⟶ H of G onto a nano k-sp. H, then G is a nano k-

sp. 

Proof: Let us consider spaces kG and map kf : kG ⟶ kH.Since H is a nano k-sp., we have H = 

kH and kf = foǤg. From Theorem 3.13, it follows that kf is a nano-perf-map and this along with 

Proposition 3.14, implies that Ǥg is nano-perf. As Ǥg is a one-to-one map, it is a nano-

homeomorphism; therefore G is a nano k-sp. 

 

4. Conclusion 

  The main purpose of this paper is to present and study images of nano perfect mappings, as well 

as to present invers images of nano perfect mappings.  

 

References 

1. Thivagar, M. L.; Richard, C. On Nano Continuity. Mathematical Theory and Modelling. 2013, 

3, 7, 32–37. 

2. Yousif, Y. Y.; Hussain, L. A. Fibrewise IJ-Perfect Bitopological Spaces. Journal of Physics: 

Conference Series, IOP Publishing, Ibn Al-Haitham 1^stInternational Scientific Conference. 

December 2017, 2018, Volume 1003, 13-14, , doi :10.1088/1742-6596/1003/1/012063, pp. 1-

12. 

3. Ashaea, G. S. ;Yousif, Y.Y. Weak and Strong Forms of 𝜔-Perfect Mappings. Iraqi Journal of 

Science. 2020, Special Issue, 45-55. 

4. Majeed, R. N. Rα-Compactness on Bitopological Spases. Journal of Al-Nahrain University 

Science. 2010, Vol.13 (3), September,134-137. 

5. Mohammed, N. F.; Gasim, S. G.; Mohammed, A. S. On Cohomology Groups of Four-

Dimensional Nilpotent Associative Algebras. Baghdad Science Journal. University of 

Baghdad – Collage of Science for Woman. Published Online First: September, DOI: 

http://dx.doi.org/10.21123/bsj.2022.19.2.029, 2021, Vol. 19, No.2, PP.329-335, 202. 

6. Nasef, A. A.; Aggour, A. I.; Aggour, S. M. On some classes of nearly open sets in nano 

topological spaces. Journal of the Egyptian Mathematical Society. 2016, 24,585-589. 

7. Vadivel, A. ; Padma, A. ; Saraswathi, M.; Saravanakumar, G. Nano Continuous Mappings via 

Nano ℳ Open Sets. Applications and Applied Mathematics : An International Journal (AAM). 

December 2021, ISSN : 1932-9466, Issue 2 , Vol.16,  1099-1119,. 

8. Engelking, R. General Topology. Heldermann. 1989, Berlin. 

http://dx.doi.org/10.21123/bsj.2022.19.2.029


IHJPAS. 36 (3) 2023 

407 
 

9. Mohammed, N. F. On Semi 𝛼-connected  Subspaces. Baghdad Science Journal. University of 

Baghdad – Collage of Science for Woman.  2010, Vol. 7, No.1, PP.174-179. 

10. Majeed, R. N. Compactness on Bitopological Spases. Journal of Al-Qadisiyah for Computer 

Science and Mathematics. 2011, Vol. 3(1), pp. 1-7. 

11. Thivagar, M. L. ; Richard, C. On nano continuity. International Journal of Mathematics and 

statistics invention. 2013,  1(1), 31–37. 

12. Parimala, M. ; Indirani, C. ; Jafari, S. ON NANO b – OPEN SETS IN NANO TOPOLOGICAL   

SPACES. Jordan Journal of Mathematics and Statistics (JJMS) 9(3). 2016  pp 173-184. 

13. Rajasekaran, I. ; Nethaji, O. Simple Forms of Nano Open Sets in an Ideal Nano Topological 

Spaces. Journal of New Theory.  2018,  ISSN: 2149-1402, Number: 24, pages: 35-43.  

14. El-Atik, A. A. ; Hassan, H. Z. Some nano topological structures via ideals and graphs. Journal 

of the Egyptian Mathematical Society, 2020. 

15. Krishnaprakash, S. ; Ramesh, R. ; Suresh, R. Nano Compactness and Nano Connectedness in 

Nano Topological space. International Journal of Pure and Applied Mathematics. 2018, Vol. 

119, no.13, pp. 107 –115. 

16. Bourbaki, N . General Topology; Part Ι. Addison Wesley. Reading. Mass, 1996. 

17. Upadhya, A. C. ; Mamata, M. k. ON NANO GP-REGULAR AND NANO GP-NORMAL 

SPACES. The International journal of analytical and experimental modal analysis. 

Karnataka. India. July, 2020, ISSN NO:0886-9367, p:3770-3775. 

18. Pears, A. R. Dimension Theorem of General Spaces. Cambridge University Press. London. 

New York, Melbourne, 1975. 

19. Ashaea, G. S. ; Yousif, Y. Y. Some Types of Mappings in Bitopological Spaces. Baghdad 

Science Journal. University of Baghdad – Collage of Science for Woman, Published Online 

First: December 2020, , DOI: http://dx.doi.org/10.21123/bsj.2020.18.1.0149, 2021, 18,1, 149-

155. 

20. Noori, S. ; Yousif, Y. Y. Soft Simply Compact Spaces. Iraqi Journal of Science. University of 

Baghdad, The 1st Conferences of Mathematics Held at Mustansiriyah University – College of 

Basic Education in 5-6 Feb. 2020, DOI:10.24996/ijs.2020.SI.1.14, Special Issue,108-113. 

21. Esmaeel, R.B. ; Saeed, S. Nano ag-open set. Journal of Physics: Conference Seriesthis link is 

disabled. 2021, 1897(1), 012031. 

22. Esmaeel,R. B.; Jassam, A. A.  On Nano ḟ-pre-g-Open Set. Ibn Al-Haitham Journal for Pure 

and Applied Science. 2020, 33(3),1-12.