IHJPAS. 36 (4) 2023 

396 

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

 

   

 

 
 

 

 

 

 

 

 

*Corresponding Author: maged.hameed1203a@ihcoedu.uobaghdad.edu.iq 

 

 
  

 

 

Abstract 

The essential objective of this paper is to introduce new notions of fibrewise topological spaces 

on D that are named to be upper perfect topological spaces, lower perfect topological spaces, 

multi-perfect topological spaces, fibrewise upper perfect topological spaces, and fibrewise lower 

perfect topological spaces. fibrewise multi-perfect topological spaces, filter base, contact point, 

rigid, multi-rigid, multi-rigid, fibrewise upper weakly closed, fibrewise lower weakly closed, 

fibrewise multi-weakly closed, set, almost upper perfect, almost lower perfect, almost multi-

perfect, fibrewise almost upper perfect, fibrewise almost lower perfect, fibrewise almost multi-

perfect, upper* continuous fibrewise upper∗ topological spaces respectively, lower* continuous 

fibrewise lower∗ topological spaces respectively, multi*-continuous fibrewise multi∗-topological 

spaces respectively multi-Te, locally In addition, we find and prove several propositions linked 

to these notions. 

 

Keywords: Fibrewise topological spaces, filter base, fibrewise upper perfect topological spaces, 

fibrewise lower perfect topological spaces, and fibrewise multi-perfect topological spaces. 

  

1. Introduction  

We begin our work with the concept of category of Fibrewise (briefly, 𝔽.𝕎.) set on a known set, 

named the base set. If the base set is stated with D, then a F.W. set on D applied to a set E with a 

doi.org/10.30526/36.4.2911 

Article history: Received 21 June 2022, Accepted 9 October 2022, Published in October 2023 

 

 

 

Ibn Al-Haitham Journal for Pure and Applied Sciences 

Journal homepage: jih.uobaghdad.edu.iq 

 

Fibrewise Multi-Perfect Topological Spaces 

                   Majed .H. J𝐚ber *                                                                    

Department of Mathematics, College of Education 

for Pure Sciences, Ibn Al-Haitham 

University of Baghdad, Iraq 

 

 

Y𝐨us𝐢f. Y. Y𝐨us𝐢f  

Department of Mathematics, College of Education 

for Pure Sciences, Ibn Al-Haitham 

University of Baghdad, Iraq 

 

                   M. El Sayed                                                                    

Department of Mathematics, College of Science 

and Arts, Najran University, Kingdom of Saudi 

Arabia 

 

https://creativecommons.org/licenses/by/4.0/
mailto:maged.hameed1203a@ihcoedu.uobaghdad.edu.iq
mailto:maged.hameed1203a@ihcoedu.uobaghdad.edu.iq
mailto:maged.hameed1203a@ihcoedu.uobaghdad.edu.iq
mailto:yoyayousif@yahoo.com
mailto:mebadria@nu.edu.sa


IHJPAS. 36 (4) 2023 

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function X is X: E → D, named the projection (briefly, project). For every point d of D, the fiber 

on d is the subset Ed = X−1(d) of E; fibers will be empty, so we do not require X to be a surjection. 

Also, for every subset D* of D, we regard ED∗ = X−1(D∗) as a 𝔽.𝕎. set on D* with the project 

determined by X. A multi-function [2] Ω of a set E into F is a correspondence such that Ω (e) is 

a nonempty subset of F for every e ∈ E. We will denote such a multi- function by Ω: E → F . For 

a multi- function Ω, the upper and lower inverse set of a set K of F, will be denoted by Ω+(K) 

and Ω−(K), respectively, that is Ω+(K) = {e ∈ E : Ω(e) ⊆ K} and Ω−(K) = {e ∈ E: Ω(e) ∩ K ≠

∅ }. 

 

Definition 1.1. [7] Suppose that E and F are 𝔽.𝕎. sets on D, with project. 𝑋𝐸: 𝐸 →  𝐷 and 

𝑋𝐹: 𝐹 →  𝐷, respectively, a function Ω: E → F is named to be 𝔽.𝕎. if 𝑋𝐹𝛰Ω =  𝑋𝐸, that is to 

say if Ω(Xd) ⊂ Fd for every point d of D. 

For other concepts or information that are undefined here, we follow nearly [3] and[4] 

Recall that [7] Let D be a topological space, the 𝔽.𝕎. Topology space (briefly, 𝔽.𝕎.T.S.) on a 

𝔽.𝕎. set E on D, which means any topology on E for that the project  X is continuous. 

Remark 1.1. [7]  

i. The smaller topology is the topology trace with X, where in the open sets of E are the pre 

image of the open sets of D, this is named the 𝔽.𝕎. indiscrete topology. 

ii. The 𝔽.𝕎.T.S. on D is stated to be a 𝔽.𝕎. set on D with a 𝔽.𝕎.T.S. 

     We regard the topology product D × T, for any topological space T, as a 𝔽.𝕎.T.S. on D using 

the category of the first projection. The equivalences in the category of 𝔽.𝕎.T.S. are named 

𝔽.𝕎.T. equivalences. If E is 𝔽.𝕎.T. equivalent to D × T, for some topological space T, we say 

that E is trivial, as a 𝔽.𝕎.T.S. on D. In 𝔽.𝕎.T. the form neighbourhood (briefly, 𝜂ℙ𝕕) is used in 

the same sense as it is in normally topology, but the forms 𝔽.𝕎. basic may need some illustration, 

so let E be 𝔽.𝕎.T.S. on D, if e is a point of Ed where in d ∈ D, appear a family N(e) of 𝜂ℙ𝕕 of e 

in E as 𝔽.𝕎. basic if as every 𝜂ℙ𝕕 H of e we have Ew ∩ K ⊂ H, for some element K of N(e) and 

𝜂ℙ𝕕 W of d in D. As example, in the case of the topological product D × T, where in T is a 

topological spaces, the family of Cartesian products D × N(t), where in N(t) runs through the 

𝜂ℙ𝕕𝑠 of t, is 𝔽.𝕎. basic for (d, t). 

𝐃𝐞𝐟𝐢𝐧𝐢𝐭𝐢𝐨𝐧 1.2. [7] The 𝔽.𝕎. functions Ω: E→ F; E and F are 𝔽.𝕎. spaces on D is named: 

(a)  Continuous (briefly, cont.) if every e ∈ Ed; d ∈ D, the inverse image of every open set of Ω(e)  

is an open set of e. 

(b)  Open if for every e∈E_d, d ∈D, the direct image of every open set of e is an open set of Ω(e). 

Definition 1.3. [7] The F.W.T.S. E on D is named 𝔽.𝕎. closed (resp., open) if the project. X is 

closed (resp., open) functions. 

Definition 1.4. [1] Let Ω: E → F be a multi-function. Then Ω is upper cont. (briefly, U. cont.) if 

Ω+ (K) open in E for all K open in F. That is, Ω+ (K) = {x ∈ E: Ω(x) ⊆ K}. K ⊆ F. 

Definition 1.5. [1] Let Ω: E → F be a multi-function. Then Ω is lower cont. (briefly, L. cont.) if 

Ω- (K) open in E for all K open in F. That is, Ω-(K) = {e ∈ E: Ω(e) ∩ K ≠∅ }. K ⊆ F 

Let Ω: E → F be a multi-function. Then Ω is multi cont. (briefly, M. cont.) if it is U. cont. and L. 

cont.  

Definition 1.6.[5] Let D be topological space, the 𝔽.𝕎. upper topology space (briefly, 

𝔽.𝕎.U.T.S.) on a 𝔽.𝕎. set E on D mean any topology on E for which the project. X is U. cont. 

 



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Definition 1.7.[5] Let D be topological space the 𝔽.𝕎. lower topology space (briefly, 

𝔽.𝕎.L.T.S.) on a 𝔽.𝕎. set E on D mean any topology on E for which the project. X is L. cont. 

Let D be topological space the 𝔽.𝕎.  multi-topology  space (briefly, 𝔽.𝕎.M.T.S.)  if  it is 

𝔽.𝕎.U.T.S. and 𝔽.𝕎.L.T.S. 

 

Definition 1.8. [3] A filt𝑒r ℑ on topological space (E,τ) a non-empty collection of non-empty 

subsets of E such that 

i. ∀ 𝔽1, 𝔽 2 ∈ ℑ, 𝔽 1 ∩ 𝔽 2 ∈ ℑ 

ii. If 𝔽 1 ⊆ 𝔽 2 ⊆E and 𝔽1∈ ℑ then 𝔽2
 ∈ ℑ. 

Definition 1.9. [3] If ℑ,𝔔 filter bases on (E,𝜏), we namely 𝔔 is fin𝑒r than ℑ (writt𝑒n as ℑ < 𝔔) 

if for all 𝔽 ∈ ℑ, there is G ⊆ 𝔽 meets𝔔 if 𝔽∩G ≠ ∅ for 𝑒v𝑒ry 𝔽 ∈ ℑ and G ∈ 𝔔. 

Definition 1.10. [10] If E is topological space and e ∈ E a 𝜂ℙ𝕕 of e is a s𝑒t 𝔘 which contain an 

op𝑒n s𝑒t V containing e. If 𝒜 is op𝑒n s𝑒t and contains e w𝑒 nam𝑒ly 𝒜 is op𝑒n 𝑎 𝜂ℙ𝕕 for a point 

e. 

Definition 1.11. [9] A p𝑜int e in (E,𝜏) is nam𝑒d t𝑜 b𝑒 a contact point of a subs𝑒t 𝒜 ⊆ E ff ∀ 𝔘 

open 𝜂ℙ𝕕 of e, cl (𝔘) ∩ 𝒜 ≠ ∅. So, s𝑒t of all c𝑜ntact p𝑜ints of 𝒜 is nam𝑒d t𝑜 b𝑒 th𝑒 closure of 

𝒜 and is symboliz𝑒d by cl (𝒜). 

Definition 1.12. [10] A subs𝑒t 𝒜 in topological space𝑒 (E,𝜏). S𝑜,  𝒜 is nam𝑒d to be 𝔼.s𝑒t in E 

(bri𝑒fly, E -s𝑒t) if ∀𝜏 an op𝑒n cov𝑒r 𝑜f 𝒜 th𝑒r𝑒 is a finit𝑒 sub coll𝑒ction H of 𝛿; 𝒜 ⊂∪{cl(H) : 

H ∈ 𝛿 }. If 𝒜 = E; then, E is named to be a OHC spac𝑒. 

Definition 1.13. [2] L𝑒t e a point in a 𝔽.𝕎.T.S. (E,𝜏) on (D,𝜌) is nam𝑒d to b𝑒 adh𝑒r𝑒nt point 𝑜f 

a F∗.B∗. ℑ. on E (bri𝑒fly, ad(e)) 𝑖ff all number of ℑ is contract a point. A set of all adherent point 

of ℑ is nam𝑒d to b𝑒 th𝑒 adh𝑒r𝑒nc𝑒 of ℑ and is symboliz𝑒s by ad(ℑ). 

Definition 1.14.[11] Th𝑒 filt𝑒r bas𝑒 ℑ (bri𝑒fly F∗.B∗. ℑ) 𝑜n t𝑜p𝑜l𝑜gical spac𝑒 (E,𝜏) is nam𝑒d t𝑜 

b𝑒 c𝑜nv𝑒rg𝑒nt (bri𝑒fly, c𝑜nv.) (Written, ℑ −−conv.→ e 𝑖ff 𝑒v𝑒ry 𝜏.op𝑒n. 𝜂ℙ𝕕 𝔘 of e, c𝑜ntains 

s𝑜m𝑒 𝑒l𝑒m𝑒nts of ℑ. 

Definition 1.15.[11] The F∗.B∗. ℑ on topological spac𝑒 (E,𝜏) is nam𝑒d dir𝑒ct𝑒d t𝑜ward a s𝑒t 𝒜 

⊂ E,(briefly, ℑ −− d.t→ 𝒜) 𝑖ff all F∗.B∗.𝔔. larg𝑒r than ℑ has an  adh𝑒r𝑒nt p𝑜int in 𝒜, i.e. ad(𝔔) 

∩𝒜 ≠ ∅, and in anoth𝑒r writing ℑ −ad→ e t𝑜 imply that ℑ −−d.t→{e}, in which e ∈E. 

Currently, we review a characterization of a point e 𝑜f a F∗.B∗. ℑ. 

 

2. Fibrewise Multi-Perfect Topological Spaces 

    In this segment we establish F.W. multi-perfect topological spaces (briefly, 𝔽.𝕎.M.P.T.S.), 

and confirmation of few of its basic characteristics. 

Definition 2.1.  Let Ω : (E,𝜏) → (F,𝜎) be a function where  E and F are 𝔽.𝕎.T.S. on D is named 

to be upper perfect (briefly, U.P.) if for every F∗.B∗. ℑ on Ω(E), such that ℑ .d.t., some subset 𝒜 

of Ω(E), the F∗.B∗  Ω+(ℑ) is  𝑑. 𝑡.  Ω−1(𝒜) in 𝐸. 

Definition 2.2.  Let Ω : (E,𝜏) → (F,𝜎) be a function where  E and F are 𝔽.𝕎.T.S. on D is named 

to be lower perfect (briefly, L.P.) if for every F∗.B∗. ℑ on Ω(E), such that ℑ .d.t., some subset 𝒜 

of Ω(E), the F∗.B∗  Ω−(ℑ) is  . 𝑑. 𝑡. Ω−1(𝒜) in 𝐸.  

     Let Ω : (E,𝜏) → (F,𝜎) be a function where  E and F are 𝔽. 𝕎.T.S. on D is named to be multi-

perfect (briefly, M.P.) if it i𝑠 U.P. and L.P. 

Lemma 2.1. A function  Ω  : (E,𝜏) → (F,𝜎) is closed if cl(Ω(𝒜)) ⊂ Ω(cl(𝒜)) for every 𝒜 ⊂ E. 



IHJPAS. 36 (4) 2023 

399 
 

Proof. (⇒) Let Ω be closed and 𝒜 ⊂ H. Since Ω is closed then Ω (cl(𝒜)) is closed set in F, 

because  cl(𝒜)  is closed set in E. so,  cl(Ω(𝒜)) ⊂  Ω (cl(𝒜)). 

(⇒) Let A be closed set in E, so 𝒜 = cl(𝒜), however cl(Ω (𝒜)) ⊂ Ω (cl(𝒜)), so cl(Ω (𝒜)) ⊂ Ω 

(𝒜). Then, Ω (𝒜) is closed in F. Therefore Ω is closed.  

Lemma 2.2. The point e in topological space (E,τ) is an ad point of a F∗.B∗. ℑ on E if  ∃ 𝑎 

F*.B*. ℑ. larger than ℑ such that ℑ ∗ −−conv.→ e. 

Proof. (⇒)Assume that e is an ad point of a F∗.B∗. ℑ. on E, then it is an C. point of every number 

of ℑ. This returns, for each 𝜏-open 𝜂ℙ𝕕 𝔘 of h, we have cl(𝔘)∩ 𝔽 ≠ ∅ for every number 𝔽 in ℑ. 

Consequently, cl(𝔘) contains a some member of any F∗.B∗. ℑ ∗ large𝑟 than ℑ such that  ℑ ∗ 

−−−conv.→ e. 

(⇐) Assume that e is not an ad point of a F∗.B∗. ℑ. on E, then ∃ 𝔽 ∈ ℑ such that e is not an contact 

of 𝔽. So, ∃ 𝜏− open- 𝜂ℙ𝕕 𝔘 of e such that cl(𝔘) ∩ 𝔽 = ∅. Denote by ℑ ∗ the family of sets 𝔽 ∗ = 

𝔽 ∩ cl(𝔘) for 𝔽 ∈ ℑ, so the sets in which 𝔽 ∗ ≠ ∅. Additionally, is a . and really is 𝔽 ∗ from 

ℑ. This is, given 𝔽1
∗  = 𝔽1 ∩ (𝐸 ∖ 𝑐𝑙(𝔘))  and  𝔽2

∗  = 𝔽2 ∩ (𝐸 ∖ 𝑐𝑙(𝔘)), ∃ 𝔽3 = 𝔽1 ∩ 𝔽2, and this 

gives  𝔽2
∗  = 𝔽3 ∩ (𝐸 ∖ 𝑐𝑙(𝔘)) ⊂  𝔽1 ∩ 𝔽2 ∩ (𝐸 ∖ 𝑐𝑙(𝔘)) =  𝔽1 ∩ (𝐸 ∖ 𝑐𝑙(𝔘)) ∩ 𝔽2 ∩ (𝐸 ∖

𝑐𝑙(𝔘)). Since F∗ is not conv. to e. So, lead to a C!!!, and h is an ad point of a F∗.B∗. ℑ. on E.  

Lemma 2.3. Assume that ℑ  is a F∗.B∗. ℑ on a topological space (E,𝜏). Suppose that e ∈ E, so ℑ 

−−conv.→ e if ℑ −− d.t→e. 

Proof . (⇐) If ℑ does not conv. to e, then, ∃𝜏-open 𝜂ℙ𝕕 𝔘 of e such that cl(𝔘)) ⊄ 𝔽 = ∅ for every 

𝔽 ∈ ℑ. Then, 𝔔 = {cl(𝔘) ∩ 𝔽 : 𝔽 ∈ ℑ } is a ℑ be a F∗.B∗. ℑ. on E larger than ℑ, and e ∉ ad of 𝔔. 

Thus, ℑ cannot be d.t. e, so lead to a then C!!!,. Then, ℑ is conv. to e. (⇒). It is clear  

Definition 2.3. The 𝔽. 𝕎.T.S. (E,𝜏) on a topological space (D,𝜌) is named to be 𝔽. 𝕎. upper 

perfect (briefly, 𝔽. 𝕎.U.P.) if the projection X is U.P. 

Definition 2.4. The F.W.T.S. (E,τ) on topological space (D,ρ) is named to be 𝔽. 𝕎. lower perfect 

(briefly, 𝔽.𝕎..L.P.) if the projection X is L.P. 

The F.W.T.S. (E, τ) on topological space (D,ρ) is named to be 𝔽. 𝕎. multi-perfect (briefly, 

𝔽. 𝕎.M.P.) if it is 𝔽. 𝕎.U.P. and 𝔽. 𝕎.L.P. 

 In the next theory we prove that just points of D can be enough for the subset A in Definition 

(1.15), and so direction. Since converge can be replaced in view of Lemma (2.2.) 

 

Theorem 2.1. Assume that (E,𝜏) is a 𝔽. 𝕎.T.S. on a topological space (D,ρ). So, the next are 

equivalent: 

i. (E,𝜏) is 𝔽. 𝕎.U.P.T.S. (resp., 𝔽. 𝕎.L.P.T.S.). 

ii. F∗.B∗. ℑ 𝑜n X(E), where conv. to a point d in D,𝐸ℑ
+ −−d.t→  Ed(resp., 𝐸ℑ

−−−d.t→ Ed). 

iii. ∀ F∗.B∗. ℑ on E, ad X(ℑ) ⊂ X(ad ℑ) 

Proof . (i) ⇒(ii) By Lemma 2.2. 

(ii) ⇒(iii) Assume that d ∈ ad X(ℑ). Thereafter, by Lemma (2.2.), ∃ F∗.B∗. 𝔔 on X(E) larger 

from X(ℑ).s.t 𝔔 –conv.→ d. Let 𝔘 = {𝐸𝔔 ∩ ℑ : G ∈ 𝔔 and 𝔽 ∈ ℑ } Thereafter, 𝔘 is a F∗.B∗. on 

E larger from 𝐸𝔔. Since 𝔔 −−d.t.→ d, by Lemma (2.3.) and X 𝑖𝑠 P., 𝐸𝔔
+−−d.t.→ Ed(resp., 

𝐸𝔔
−−−d.t.→ Ed). 𝔘 being larger than 𝐸𝔔 , we have Ed ∩ Ω+(ad 𝔘) ≠ ∅(resp. , 𝐸𝑑 ∩ Ω−(𝑎𝑑 𝔘)  ≠

∅. ). Hence it is obvious that EdΩ(ℑ ) ≠ ∅. So, d ∈ X(ad ℑ). 

(iii)⇒(i) Let ℑ be a F∗.B∗. on X(E) such that it is d.t. some subset 𝒜 of X(E). Assume that 𝔔 is 

a F∗.B∗. on E larger than 𝐸ℑ. Thereafter, X(𝔔) is a F∗.B∗. on X(E) larger than ℑ and so 𝒜 ∩ (ad 



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X(𝔔)) ≠ ∅. Then, by (c), 𝒜 ∩ X(ad (𝔔)) ≠ ∅ such that 𝐸𝒜
+  ∩ (ad (𝔔)) ≠ ∅(resp. , 𝐸𝒜

−  ∩ (ad (𝔔)) 

≠ ∅). Then, 𝐸ℑ is d.t. 𝐸𝒜. So, X is U.P.(resp., L.P.). 

 

Corollary 2.1. Assume that (E,τ) is a F.W.T.S. on a topological space (D,ρ). So, the next are 

equivalent: 

i. (E,𝜏)  is 𝔽. 𝕎.M.P.T.S.. 

ii. F∗.B∗. ℑ 𝑜n X(E), where  conv. to a point d in D,𝐸ℑ
+ −−d.t→  Ed(resp., 𝐸ℑ

− −−d.t→ Ed). 

iii. ∀ F∗.B∗. ℑ on E,ad X(ℑ) ⊂ X(ad ℑ) 

Theorem 2.2. If the 𝔽. 𝕎.T.S. (E,𝜏) on (D,𝜌) is U.P.(resp., L.P.), then it is  closed. 

Proof. Suppose that E is a 𝔽. 𝕎.U.P.T.S. (resp., 𝔽. 𝕎.L.P.T.S.) on D, then the projection XE: E 

→ D is U.P. (resp., L.P.) to show that it is closed, by Theorem (4.1.16.) (a)  ⇒ (c) for any F∗.B∗. ℑ 

on E ad X(ℑ) ⊂ X(ad (D)), by  Lemma (4.1.11.), Ω is  closed if cl(Ω(𝒜)) ⊂ (cl(𝒜)) for every 𝒜 

⊂E, so X is closed in which ℑ = { 𝒜 }.  

Corollary 2.2. If the 𝔽. 𝕎.T.S. (E,𝜏) on (D,𝜌) is M.P., then it is closed. 

 

3. Fibrewise Multi-Perfect and multi-Rigidity Topological Spaces. 

     In this segment, we present the idea of multi-perfect topological, upper rigidity spaces  lower 

rigidity spaces,  multi-rigidity spaces  and make sure of some of its base characteristics. 

 

Definition 3.1. A subset 𝒜 of a topological space (E,τ) is named to be upper rigid in E (briefly, 

U.R.) if for every F∗.B∗. ℑ on E 𝑎𝑑 𝑋+(ℑ) ∩ 𝒜 =  ∅, ∃𝔘 ∈  𝜏 and 𝔽 ∈ ℑ such that 𝒜 ⊂  𝔘  or 

equivalently, if for every F∗.B∗. ℑ on E, whenever 𝒜 ∩ (ad ℑ) = ∅, thereafter for some F ∈ ℑ, 

𝒜 ∩ (cl(ℑ)) = ∅. 

Definition 4.2. A subset 𝒜 of topological space (E,τ) is named to be lower rigid in E (briefly, 

L.R.) if for every F∗.B∗. ℑ on E 𝑎𝑑 𝑋−(ℑ) ∩ 𝒜 =  ∅, ∃𝔘 ∈  𝜏 and 𝔽 ∈ ℑ such that 𝒜 ⊂  𝔘  or 

equivalently, if for every F∗.B∗. ℑ on E, whenever 𝒜 ∩ (ad ℑ) = ∅, thereafter for some F ∈ ℑ, 

𝒜 ∩ (cl(ℑ)) = ∅.  

 A subset 𝒜 of topological space (E,τ) is named to be multi-rigid in E (briefly, M.R.) if it is U.R. 

and L.R. 

Theorem 3.1. If (E,𝜏) is a 𝔽. 𝕎. closed topological space on (D,ρ) such that every 𝐸𝑑
+(resp. , 𝐸𝑑

−). 

in which d ∈ D is U.R.(resp., L.R.) in E, then (E,𝜏) is a 𝔽. 𝕎.U.P. (resp., 𝔽. 𝕎.L.P.). 

Proof. Suppose that E is a 𝔽. 𝕎. closed topological space on D, thereafter 𝑋𝐸 : E →D exists T.P. 

it is U.P.(resp., L.P.), assume that ℑ is a F∗.B∗. on 𝑋𝐸such that D –conv.→ d in D, for some d in 

D. If 𝔔 is a F∗.B∗ on E larger than the F∗.B∗.𝐸ℑ, then 𝑋(𝔔) is a F∗.B∗. on D, larger than ℑ. 

Because ℑ −−d.t.→ d by Lemma (2.3.), d ∈ adX(𝔔), i.e, d ∈ ∩{ad X(G;G ∈ 𝔔)}, and hence, d ∈ 

∩{X(ad G;G ∈ 𝔔)} by Lemma 1.1.). By X 𝑖s closed, so 𝐸𝑑
+ ∩ ad (G) ≠ ∅(resp., 𝐸𝑑

− ∩ ad (G) 

≠ ∅), for every G ∈ 𝔔. So, for every 𝔘 ∈ 𝜏 𝑤𝑖th 𝐸𝑑
+(resp. , 𝐸𝑑

−)⊂ 𝔘, cl(𝔘) ∩G ≠ ∅  for every G 

∈ 𝔔. Since, 𝐸𝑑
+(resp. , 𝐸𝑑

−) is U.R.(resp., L.R.), it then follows that 𝐸𝑑
+∩ ad (𝔔) ≠ ∅(resp. , 𝐸𝑑

−∩ 

ad (𝔔) ≠ ∅) . Thus, 𝐸ℑ −−d.t.→Ed. S𝑜 by Theorem [(2.1.), (b) ⇒(a)], X is U.P.(resp., L.R.)  

Corollary 3.1. If (E,𝜏) is a 𝔽. 𝕎. closed topological space on (D,𝜌) such that every Ed in which 

d ∈ D 𝑖𝑠 M.R. in E, then (E,𝜏) is a 𝔽. 𝕎.M.P. 

Theor𝒆m 3.2. If the 𝔽. 𝕎.T.S.  (E,𝜏) on (D,𝜌) is U.P. (resp., L.P.), then, it is closed and for every 

d ∈ B, 𝐸𝑑
+(resp. , 𝐸𝑑

−). is U.R.(resp., L.R.) in E. 



IHJPAS. 36 (4) 2023 

401 
 

Proof. Let E be a 𝔽. 𝕎.T.S. on D, so the projection 𝑋𝐸 : E→ D exists and it is U. cont.(resp., L. 

cont.).  𝑋𝐸 is an U.P.(resp., L.P.) so it is closed. T.P. is closed and for every d ∈ D,

𝐸𝑑
+(resp. , 𝐸𝑑

−) is U.R.(L.R) in E. Le𝑡 d ∈ D and suppose  ℑ is a ℑ ∗.B*. on E such that 

(ad ℑ)∩ 𝐸𝑑
+= ∅(resp. , (ad ℑ) ∩ 𝐸𝑑

−  =  ∅). Therefore, d ∉ 𝑋𝐸 (ad ℑ) By 𝑋𝐸 is U.P. (resp., L.P.), 

by Theorem [(2.1.) (a) ⇒ c)], d ∉ ad𝑋𝐸 (ℑ). Thus, ∃ an 𝔽∈ ℑ such that d ∉ ad𝑋𝐸 (𝔽).∃an 𝜌−open 

a 𝜂ℙ𝕕 V of d such that cl(V)∩ 𝑋𝐸 (𝔽) = ∅. Since 𝑋𝐸 is cont., for every e ∈𝐸𝑑
+(resp. , 𝐸𝑑

−). We 

shall get a 𝜏-open a 𝜂ℙ𝕕 𝔘e of e such that 𝑋𝐸 (cl(𝔘e)) ⊂ cl(V) ⊂ D − 𝑋𝐸 (𝔽). So 𝑋𝐸 (cl(𝔘e)) ∩ 

𝑋𝐸 (𝔽) = ∅, so that cl(𝔘e)) ∩ 𝔽 = ∅. Then h ∉ cl(𝔽), for every e ∈ 𝐸𝑑
+(resp. , 𝐸𝑑

−), so 𝐸𝑑
+(resp. , 𝐸𝑑

−) 

∩ cl(𝔽) = ∅, So 𝐸𝑑
+(resp. , 𝐸𝑑

−) is U.R.(resp., L.R.) in E.   

Corollary 3.2. If the 𝔽. 𝕎.T.S.  (E,𝜏) on (D,𝜌) is M.P. then it is  closed and for every d ∈ B, Ed 

is M.R. in E. 

Definition 3.3. The function Ω : (E,𝜏) → (F,𝜎) is named to be weakly upper closed (briefly, W.U. 

closed) if ∀f ∈ Ω+(E) and ∀ 𝔘 ∈ 𝜏 containing η−1(f) in E,∃ a 𝜌 −open a 𝜂ℙ𝕕 V of d such that 

Ω−1(V) ⊂ cl(𝔘). 

Definition 3.4. The function Ω : (E,𝜏) → (F,𝜎) is named to be weakly lower closed (briefly, W.L. 

closed) if ∀f ∈ Ω−(E) and ∀ 𝔘 ∈ 𝜏 containing Ω−1(f) in E,∃ a 𝜌 −open a 𝜂ℙ𝕕 V 𝑜f d such that 

Ω−1(V) ⊂ cl(𝔘). 

 The function Ω : (E,𝜏) → (F,𝜎) is named to be weakly multi-closed (briefly, W.M. closed) if it 

is W.U. closed and W.L. closed.  

 

Definition 3.5. The 𝔽. 𝕎.T.S. (E,𝜏) on (D,𝜌) is named to be 𝔽. 𝕎. upper weakly closed (briefly, 

𝔽. 𝕎.U.W. closed) if the projection X is W.U. closed.  

 

Definition 3.6. The 𝔽. 𝕎.T.S. (E,𝜏) on (D,𝜌) is named to be 𝔽. 𝕎. lower weakly closed (briefly, 

𝔽. 𝕎.L.W. closed) if the projection X is W.L. closed. 

The 𝔽. 𝕎.T.S. (E,𝜏) on (D,𝜌) is named to be 𝔽. 𝕎. multi-weakly closed (briefly, 𝔽. 𝕎.M.W. 

closed) if it is 𝔽. 𝕎.U.W. closed and 𝔽. 𝕎.U.W. closed. 

 

Theorem 3.3. The 𝔽. 𝕎. closed topological space (E,𝜏) on (D,𝜌) is W.U. closed (resp., W.L. 

closed). 

Proof. Assume that E is a 𝔽. 𝕎. closed topological space on D, then the projection XE: E→ D 

exists, and to prove its W.U. closed (resp., W.L. closed). Let d ∈ 𝑋𝐸 and le𝑡 𝔘 ∈ 𝜏 containing 

𝐸𝑑
+(resp. , 𝐸𝑑

−) in E. Currently, by Theorem (4.1.18.) cl(E−cl(𝔘)) = cl(E−cl(𝔘)), and, hence by 

Lemma, (4.1.11.) and since 𝑋𝐸 is closed, we have cl(𝑋𝐸 (E−cl(𝔘))) ⊂ 𝑋𝐸 [cl(E−cl(𝔘))].. 

Currently, since d ∉ 𝑋𝐸 [cl(E−cl(𝔘))], d ∉ cl(𝑋𝐸 (E−cl(U))), and thus, ∃an 𝜌−open a 𝜂ℙ𝕕 V of d 

∈ D such that cl(V)∩ 𝑋𝐸 (E−cl(𝔘)) = ∅ which means that 𝐸𝑐𝑙(𝑉)
+ ∩(E−cl(𝔘)) = ∅(resp.,

𝐸𝑐𝑙(𝑉)
− ∩(E−cl(𝔘)) = ∅), and so 𝑋𝐸 is W.U. closed (resp., W.L. closed).  

Corollary 3.3. The 𝔽. 𝕎. closed topological space (E,τ) on (D,ρ) is W.M. closed. 

 

Theorem 3.4. Let (E,τ) be 𝔽. 𝕎.T.S. on (D,ρ). Then (E,τ) is 𝔽. 𝕎.U.P.(resp., 𝔽. 𝕎.L.P.), if: 

i. (E,𝜏) is 𝔽. 𝕎.U.W. closed (resp., 𝔽. 𝕎.L.W. closed) topological space. 

ii. 𝐸𝑑
+(resp. , 𝐸𝑑

−) is U.R.(resp., L.R.), for every d ∈ D. 

Proof. Assume that E is a 𝔽. 𝕎. space on D satisfying the conditions (i) and (ii), then the 

projection XE: E→ D exists. To prove that XE 𝑖s U.R.(resp., L.R.), we have to show in view of 



IHJPAS. 36 (4) 2023 

402 
 

Theorem (3.1.) that XE is closed. Let d ∈ XE(𝒜), for some not empty subset 𝒜 of E, but d ∉ 

XE(cl(𝒜)). Then, E = { 𝒜 } is a F∗.B∗ 𝑜n E and (ad(E))∩ 𝐸𝑑
+(resp. , 𝐸𝑑

−) = ∅. By U.R.(resp., 

L.R.) of 𝐸𝑑
+(resp. , 𝐸𝑑

−), a ∃ 𝔘 ∈ 𝜏 containing 𝐸𝑑
+(resp. , 𝐸𝑑

−) such that cl(𝔘)∩ 𝒜 = ∅. By W.U. 

closed (resp., W.L. closed) of XE ∃ an 𝜌−open 𝑎 𝜂ℙ𝕕 D of d such that, 𝐸𝑐𝑙(𝑉) 
+ ∩ 𝒜 = 

∅(resp. , 𝐸𝑐𝑙(𝑉) 
− ∩ 𝒜 = ∅), i.e., cl(V) ∩ XE (𝒜) = ∅, which is impossible since d ∈ XE(𝒜). So Ω 

is closed.  

Corollary 3.4. Let (E,𝜏) be 𝔽. 𝕎.T.S. on (D,𝜌). Then, (E,𝜏) is 𝔽. 𝕎.M.P, if  

i. (E,𝜏) is 𝔽. 𝕎.M.W. closed topological space. 

ii.    𝐸𝑑 is M.R, for every d ∈ D.  

Lemma 3.1. [11]A subset A of a topological space (E,𝜏) is 𝔼. set if for every F∗.B∗ on ℑ on 𝒜; 

(ad(ℑ)) ∩ 𝒜 ≠ ∅. 

Theorem 3.5. If (E,𝜏) is 𝔽. 𝕎.U.P.T.S.(resp., 𝔽. 𝕎.L.P.T.S.) on (D, 𝜌) and D∗ ⊂ D is an 𝔼 set in 

D, so 𝐸𝐷∗
+ (resp. , 𝐸𝐷∗

− ) is an  𝔼 set in E. 

Proof. Suppose that E is a 𝔽. 𝕎.U.P.T.S. (resp., 𝔽. 𝕎.L.P.T.S.) on D, therefore XE: E → D exist. 

Le𝑡 ℑ be a F∗.B∗. on D∗. By D∗ is an  𝔼  set in D, D∗ ∩ ad XE(ℑ) ≠ ∅, by Lemma (3.1.). By 

Theorem [(2.1.) (i) ⇒ (iii)], D∗ ∩ XE(ad (ℑ) ≠ ∅, so 𝐸𝐷∗
+ ∩ ad (ℑ) ≠ ∅(resp. , 𝐸𝐷∗

− ∩ ad (ℑ) ≠ ∅). 

Hence, by Lemma (3.1.), 𝐸𝐷∗
+ (resp. , 𝐸𝐷∗

− ) is an  𝔼 set in E.  

Corollary 3.5. If (E,𝜏) is 𝔽. 𝕎.M.P.T.S on (D, 𝜌) and D∗ ⊂ D is an 𝔼 set in D, so 𝔼𝐷∗  is an  𝔼 set 

in E.  

Definition 3.7. The function Ω : (E,𝜏) → (F,𝜎) is named to be almost U.P. if for every E set K in 

F, Ω+(K) is an 𝔼 set in E. 

Definition 3.8. The function Ω : (E,𝜏) → (F,𝜎) is named to be almost L.P. if for every  𝔼 set K 

in F,  Ω-(K) is an 𝔼 set in E. 

    The function Ω : (E,𝜏) → (F,𝜎) is named to be almost M.P. if almost U.P. and almost L.P.  

Definition 3.9. The 𝔽. 𝕎.T.S. almost U.P. on (D,ρ) is named to be 𝔽. 𝕎. almost U.P. if the 

projection X is almost perfect. 

Definition 3.10. The 𝔽. 𝕎.T.S. almost L.P. on (D,𝜌) is named to be 𝔽. 𝕎. almost L.P. if the 

projection X is almost perfect. 

    The 𝔽. 𝕎.T.S. almost M.P. on (D,ρ) is named to be 𝔽. 𝕎. almost M.P. if it is 𝔽. 𝕎. almost U.P. 

and 𝔽. 𝕎. almost L.P. 

Theorem 3.6. Let (E,𝜏) be 𝔽. 𝕎.T.S. on (D,𝜌) such that: 

i. For every d ∈ D, 𝐸𝑑
+(resp. , 𝐸𝑑

−) is U.R.(resp., L.R.) and 

ii. (E,𝜏) be 𝔽. 𝕎.U.W. closed (resp., 𝔽. 𝕎.L.W. closed) topological space. Then, (E,τ) is 𝔽. 𝕎. 

almost U.P.T.S.(resp., 𝔽. 𝕎. almost L.P.T.S.). 

Proof. Le𝑡 E be 𝔽. 𝕎.T.S. on D, so XE : E→ D exist and it is U. cont. (resp., L. cont.). Assume 

that D∗ is an 𝔼 set in D and let ℑ be a F∗.B∗. on ED∗. Currently, XE(ℑ) is a F∗.B∗. on D∗ and so 

by Lemma (4.2.15.), (ad XE(ℑ)) ∩ D∗ ≠ ∅. Let d ∈ (ad XE(ℑ)) ∩ D∗. Let ℑ has no ad point in 

𝐸𝐷∗
+ (resp. , 𝐸𝐷∗

− ), so that (ad (ℑ))∩ 𝐸𝑑
+(resp. , 𝐸𝑑

−) = ∅. By 𝐸𝑑
+(resp. , 𝐸𝑑

−) is U.R.(resp., U.R.),∃ an 

𝔽 ∈ ℑ and 𝜏−open se𝑡 𝔘 containing 𝐸𝐷∗
+ (resp. , 𝐸𝐷∗

− ),  such that 𝔽 ∩cl(𝔘) = ∅. Since W.U. closed 

(resp., W.L. closed) of XE, ∃ 𝜌− closed a 𝜂ℙ𝕕 V of d such that E (𝜌−cl(V)) ⊂ 𝜏 − cl(𝔘) which 

means that 𝐸(𝜌−𝑐𝑙(𝑉)) 
+ ∩ 𝔽  = ∅ (resp. , 𝐸(𝜌−𝑐𝑙(𝑉)) 

− ∩ 𝔽  = ∅ ) i.e.,  𝜌 − cl(V) ∩ X(𝔽) = ∅, which is a 

contradiction. Thus, by Lemma (4.2.15.), 𝐸𝐷∗
+ (resp., 𝐸𝐷∗

− ) is an 𝔼 set in E and so XE is almost 

U.P. (resp., almost L.P.).  

Corollary 3.6. Le𝑡 (E,𝜏) be 𝔽. 𝕎.T.S. on (D,ρ) such that: 



IHJPAS. 36 (4) 2023 

403 
 

i. For every d ∈ D, Ed is M.R. and 

  (E,𝜏) be 𝔽. 𝕎.M.W. closed topological space. Then, (E,𝜏) is 𝔽. 𝕎. almost M.P.T.S. 

 

 

4. Some Result on Multi Topological Spaces 

We Currently give some results of 𝔽. 𝕎.U.P.T.S.(resp., 𝔽. 𝕎.L.P.T.S. and 𝔽. 𝕎.M.P.T.S.). The 

following characterization theorem for an U. cont. (resp., L. cont. and M. cont.) function is 

recalled to this end. 

Theorem 4.1. A topological space (E,τ) is 𝔽. 𝕎.U.T.S.(resp., 𝔽. 𝕎.L.T.S. ) on (D,ρ) if XE(cl(𝒜)) 

⊂ cl(XE(𝒜)), for each 𝒜 ⊂ E .  

Proof. (⇒) Assume that E is 𝔽. 𝕎.U.T.S.(resp., 𝔽. 𝕎.L.T.S. ) on D then the projection XE : E→ 

D exist and it is U. cont. (resp., L. cont.). Suppose that e ∈ cl(𝒜) and D is 𝜌−open a 𝜂ℙ𝕕 of Ω(e). 

Since XE is U. cont. (resp., L. cont.), ∃ an 𝜏-open a 𝜂ℙ𝕕 𝔘 of e such that X(cl(𝔘)) ⊂ cl(V). Since 

cl(𝔘) ∩ 𝒜 ≠ ∅, then cl(V) ∩ X(𝒜) ≠ ∅. So, XE(𝒜) ∈ cl(XE(𝒜)). This shows that XE(cl(𝔘)) ⊂ 

cl(XE(V)).  

(⇐) It is clear.  

Corollary 4.1. A topological space (E,𝜏) is F.W.M.T.S on (D,𝜌) if XE(cl(𝒜)) ⊂ cl(XE(𝒜)).  

Theorem 4.2. Let (E,𝜏) i𝑠 𝔽. 𝕎.U.P.T.S.(resp., 𝔽. 𝕎.L.P.T.S.) on (D,𝜌). So 𝐸𝒜
+(resp.,  

𝐸𝒜
−) preserves U.R. (resp., L.R.). 

Proof. Assume that E is a 𝔽. 𝕎.U.P.T.S.(resp., 𝔽. 𝕎.L.P.T.S.) on D, then the projection XE : E 

→ D exist and it is U. cont. (resp., L. cont.). Le𝑡 𝒜 be an U.R. set(resp.,  L.R. set ) in D and let ℑ 

be a F∗.B∗. on 𝔼 such that 𝐸𝒜∩ (ad (ℑ)) = ∅. By XE is U.R. (resp., L.R.). and 𝒜 ∩ XE(ad (ℑ)) 

= ∅, by Theorem [(2.1.)  (i)  ⇒ (iii)] we get 𝒜 ∩ (ad (𝑋𝐸( ℑ))) = ∅. Currently, a being an 

U.R.(resp., L.R.) set in D, ∃ an 𝔽∈ ℑ such that 𝒜 ∩ (cl(XE(ℑ ))) = ∅. Because XE is U. 

cont.(resp., L. cont.) and by Theorem (4.1.) it follows tℎ𝑎t 𝒜 ∩ XE(cl(ℑ)) = ∅. Tℎ𝑒n 𝐸𝒜
+∩ (cl(ℑ)) 

= ∅(𝑟𝑒𝑠𝑝. , 𝐸𝒜
−∩ (cl(ℑ)) = ∅). Tℎ𝑒n T.P. 𝐸𝒜

+(𝑟𝑒𝑠𝑝.,  𝐸𝒜
−)  is U.R.(r𝑒𝑠p., L.R.).  

W𝑒 p𝑟𝑒𝑠𝑒𝑛t tℎ𝑒 f𝑜𝑙𝑙𝑜𝑤𝑖𝑛g d𝑒𝑓𝑖𝑛𝑖𝑡𝑖𝑜n t𝑜 𝑠𝑡𝑢dy tℎ𝑒 co𝑛𝑑𝑖𝑡𝑖𝑜𝑛s u𝑛𝑑𝑒r wℎ𝑖𝑐h an F.W. al𝑚𝑜𝑠t 

p𝑒𝑟𝑓𝑒𝑐t t𝑜p𝑜𝑙𝑜gical sp𝑎𝑐𝑒 c𝑎𝑛 be an 𝔽. 𝕎.U.P.T.S.(r𝑒𝑠p., 𝔽. 𝕎.L.P.T.S.). 

 

Corollary 4.2. Let (E,𝜏) be 𝔽. 𝕎.M.P.T.S on (D,𝜌). So 𝐸𝒜 preserves M.R. 

Definition 4.1. The function Ω : (E,τ) → (F,σ) is named to be upper∗ continuous (briefly, U∗. 

cont.) if for any τ-open a 𝜂ℙ𝕕 V 𝑜f Ω+(e), ∃ an 𝜏−open a 𝜂ℙ𝕕 𝔘 of e such that  Ω(cl(𝔘)) ⊂ cl(V). 

Definition 4.2. The function Ω : (E,τ) → (F,σ) is named to be lower* continuous (briefly, L*. 

cont.) if for any τ-open a 𝜂ℙ𝕕 V of Ω−(e), ∃ an 𝜏−open a 𝜂ℙ𝕕 𝔘 of e such that  Ω(cl(𝔘)) ⊂ cl(V). 

     The function Ω : (E,τ) → (F,σ) is named to be multi∗ -cont. (briefly, M∗. cont.) if it is L∗. 

cont. and U∗. cont.  

Definition 4.3. The 𝔽. 𝕎.T.S. (E,𝜏) on (F,𝜎) is named 𝔽. 𝕎.U*.T.S. if the projection X is 

U∗.cont. 

Definition 4.4. The 𝔽. 𝕎.T.S. (E,𝜏) on (F,𝜎) is named 𝔽. 𝕎.L*.T.S. if the projection X is L∗.cont. 

     The 𝔽. 𝕎.T.S. (E,𝜏) on (F,𝜎) is named 𝔽. 𝕎.M*.T.S. if it is 𝔽. 𝕎.L*.T.S. and 𝔽. 𝕎.U*.T.S. 

Importance of the above definition for characterization of 𝔽. 𝕎.U.P.T.S.(resp., 𝔽. 𝕎.L.P.T.S. and 

𝔽. 𝕎.M.P.T.S.). It is quite clear from the next result. 

Lemma 4.1.[27] In a Urysohn topological space 𝔼 set is closed set.  

 



IHJPAS. 36 (4) 2023 

404 
 

Theorem 4.3. If (E,𝜏) is 𝔽. 𝕎.U∗.T.S.(resp., 𝔽. 𝕎.L∗.T.S.) on a Te (F,𝜎), so it is 

𝔽. 𝕎.U.P.T.S.(resp., 𝔽. 𝕎.L.P.T.S.)  if  ∀ F∗.B∗ on E, if 𝑋ℑ −−conv. →d ; d ∈ 𝐷, then ad ℑ ≠ ∅. 

Proof. (⇒) Assume that (E,τ) is a 𝔽. 𝕎.U∗.T.S.(resp., 𝔽. 𝕎.L∗.T.S.) on a Te (D,𝜌), then ∃U∗. 

cont.(resp., L∗. cont. ) projection function XE : (E,𝜏)  → (D,𝜌) and 𝑋ℑ  −−conv.→ d in which d 

∈ D, for a F∗.B∗ on ℑ  on E. So 𝐸𝑋ℑ

+  −− dir.−→𝐸𝑑
+(resp., 𝐸𝑋ℑ

−  −− dir.−→𝐸𝑑
−). By ℑ is larger than 

𝐸𝑋ℑ

+ (resp. , 𝐸𝑋ℑ

− ), 𝐸𝑑
+(resp., 𝐸𝑑

−) ∩ ad ℑ ≠ ∅, so ad ℑ ≠ ∅. 

(⇐) Assume that ∀ 𝐹 ∗. 𝐵 ∗. ℑ. 𝑜𝑛 𝐸, 𝑋ℑ  − −conv. → 𝑑  in which d ∈ D, implies  ad ℑ ≠ ∅. Let 

𝔔 be a F∗.B∗. on D such that 𝔔 –conv.→ d, and let 𝔔 ∗ be a F∗.B∗ on E, such that 𝔔 ∗ is larger 

than 𝐸𝔔. Then 𝑋𝔔∗ is larger than 𝔔. So 𝑋𝔔∗−−conv.→ d. So, ad 𝔔 ∗ ≠ ∅. Let z ∈ D such that z 

≠ d. So, by D is U.(resp., L.) Te, ∃ 𝜌−open a 𝜂ℙ𝕕 𝔘 of d and 𝜌−open a 𝜂ℙ𝕕 V of z such that (𝜌 

− cl(𝔘)) ∩ (𝜌 − cl(V)) = ∅. Since 𝑋𝔔∗ −−conv.→ d, ∃ a G ∈ 𝔔 ∗ such that XG ⊂ 𝜌 − cl(𝔘). 

Currently, by X is U∗. cont. (resp., L∗. cont.), corresponding to every e ∈ Ez, ∃ 𝜏−open a 𝜂ℙ𝕕 

W of e such that X(𝜏 − cl(V)). Thus, 𝜌 − cl(W ∩G) = ∅. It follows that 𝐸𝑍
+(resp. , 𝐸𝑍

−) ∩ 𝔔 ∗  = 

∅, ∀ z ∈ D −{d}. Consequently, 𝐸𝑑
+ ∩ ad 𝔔 ∗ ≠ ∅(resp. , 𝐸𝑑

− ∩ ad 𝔔 ∗ ≠ ∅), and X is U.P.(resp., 

L.P.) a𝑛d so (E,𝜏) is 𝔽. 𝕎.U∗.T.S.(resp., 𝔽. 𝕎.L∗.T.S.).  

C𝒐𝒓𝒐llary 4.3. If (E,𝜏) is 𝔽. 𝕎.M∗.T.S on a Te (F,𝜎), so 𝑖t is 𝔽. 𝕎.M.P.T.S if  ∀F∗.B∗ on E, if 

𝑋ℑ −−conv. →d ; d ∈ 𝐷, then ad ℑ ≠ ∅.  

 

Corollary 4.4. Let (E,τ) be 𝔽. 𝕎.M∗.T.S on (QHC) on a  Urysohn topological space (D,ρ), so 

(E,τ)  is 𝔽. 𝕎.M.T.S.. 

 

Theorem 4.4. Let (E,𝜏)  be 𝔽. 𝕎.U∗.T.S.(resp., 𝔽. 𝕎.L∗.T.S.) on locally QHC on a Te(D,𝜌), then 

(D,𝜌) is 𝔽. 𝕎.U∗.T.S.(resp., 𝔽. 𝕎.L∗.T.S.) if it is 𝔽. 𝕎. almost U.P.(resp., 𝔽. 𝕎. almost L.P.). 

Proof. (⇐) Let (E,𝜏) is 𝔽. 𝕎. almost U.P.(resp., 𝔽. 𝕎. almost L.P.), so ∃ almost U.P.(resp., almost 

L.P.) projection function XE : E → D and let D be any F∗.B∗. on E and let 𝑋ℑ −−conv.→ d in 

which d ∈ D. There are an 𝐸 set D∗ in D and 𝜌−open a 𝜂ℙ𝕕V of d such that, d ∈ V ⊆ D∗. Let E 

= {𝜌 − cl(𝔘)) ∩ 𝑋𝔽 ∩ D∗; 𝔽 ∈ ℑ and 𝔘 is a 𝜌−open a 𝜂ℙ𝕕 of d}. By L𝑒𝑚𝑚a (4.1.), D∗ is closed 

and hence no member of E is void. Reality, if not, let for some 𝜌−open a 𝜂ℙ𝕕 𝔘  of d and some 

𝔽 ∈ ℑ, 𝜌 − cl(𝔘) ∩ 𝑋𝔽 ∩D∗ = ∅. Then W = 𝔘 ∩V since d ∈ 𝔘 ∩V ∈ 𝜌 and 𝜌 − cl(W = cl(W) ⊂ 

cl(D∗) = D∗, by Lemma (4.1.). Currently ∅ = 𝜌−cl(W)∩ 𝑋𝔽∩D∗ = 𝜌 −cl(W)∩ 𝑋𝔽, which is not 

possible, since 𝑋𝔽—conv.→ d. So E is F∗.B∗. on D, and is obviously larger than 𝑋ℑ, so that E –

conv.→ d. Also 𝔔 = {𝐸𝐻
+(resp. , 𝐸𝐻

−) ∩ 𝔽 : ℋ ∈ E and 𝔽 ∈ ℑ } is obviously a filter on 

𝐸𝐷∗
+ (resp. , 𝐸𝐷∗

− ). Because X is almost U.P.(resp., almost L.P.), 𝐸𝐷∗
+ (resp. , 𝐸𝐷∗

− ) is an ℍ.set and so 

ad 𝔔 ∩ 𝐸𝐷∗
+  ≠ ∅(resp., 𝔔 ∩ 𝐸𝐷∗

−  ≠ ∅).Thus X is U.P.(resp., L.R.) and by Theorem (4.3.) (E,𝜏) 

be 𝔽. 𝕎.U∗.T.S.(resp., 𝔽. 𝕎.L∗.T.S.). 

   

C𝒐𝒓𝒐llary 4.5. Let (E,𝜏)  be 𝔽. 𝕎.M∗.T.S on locally QHC on a Te(D,𝜌), then (D,𝜌) is 

𝔽. 𝕎.M∗.T.S if it is 𝔽. 𝕎. almost M.P.. 

Lemma 4.2. [10] A topological space (E,𝜏) 𝑖𝑠 T2 ⇐⇒ {e} = cl(e) ∀ e ∈E.  

Theorem 4.5. If (E,τ)  is a 𝔽. 𝕎.U.P.(resp., 𝔽. 𝕎.U.P.) injection and surjective topological space 

with E is a U.T2 space(resp.,  L.T2 space ) on (D,𝜌), Then D is U.T2 space (resp.,  L.T2 space ). 

Proof. Let d1, d2 ∈ D such that d1 ≠ d2. By X is surjective, so d1, d2 ∈ E and p is injection, then 

𝐸𝑑1
+   ≠ 𝐸𝑑2

+ (resp., 𝐸𝑑1
−   ≠ 𝐸𝑑2

− . Since X is U.P.(resp., L.P.), so by Theorem (2.2.) it is closed. By 

Lemma (4.2.) we have {𝐸𝑑1
+ } = cl{d1} (resp., {𝐸𝑑1

− } = cl{d1)) and {𝐸𝑑2
+ } = cl{d2} (resp., {𝐸𝑑2

− } 



IHJPAS. 36 (4) 2023 

405 
 

= cl{d2}) Because X is  U.T2 space (resp., U.T2 space). Currently, X(cl{𝐸𝑑1
+ }) = cl{d1}(resp., 

X(cl{𝐸𝑑1
− }) = cl{d1}) and X(cl{𝐸𝑑2

+ }) = cl{d2}(resp., X(cl{𝐸𝑑2
− }) = cl{d2}), since X is closed. 

This mean {d1} = cl{d1} and {d2} = cl{d2}. Hence D is U.T2 space(re𝑠p.,  U.T2 space ). 

    Our following theory gives a description of an important class of 𝔽. 𝕎.U.TS.(resp., 

𝔽. 𝕎.L.TS.) meaning the QHC spaces in terms of 𝔽. 𝕎.U.P.T.S. (r𝑒𝑠p., 𝔽. 𝕎.L.P.T.S.).  

 

C𝒐𝒓𝒐llary 4.6. If (E,𝜏)  is a 𝔽. 𝕎.M.P. injection and surjective topological space with E is a 

M.T2 space on (D,𝜌), Then D is M.T2. space. 

Theorem 4.6. For a topological space (E,𝜏), the next are equivalent: 

i. H is QHC. 

ii. A 𝔽. 𝕎.U. (E,𝜏) is P.T.(resp., 𝔽. 𝕎.L. (E,𝜏) is P.T.) space with  constant projection on 

D∗ in wh𝑖𝑐h D∗ is a singleton with two equal topologies meaning the unique topology on 

D∗. 

iii. The 𝔽. 𝕎.. (B×H,Q) is U.P.T.S.(resp., L.P.T.S.) on (D,𝜌), in which 𝔔 = 𝜌 × 𝜏. 

Proof . (i) ⇒ (ii) Suppose that XE : E → D is a constant projection on D∗  where D∗ is a singleton 

with two equal topologies meaning the unique topology on D∗. X is obviously closed. 

Additionally,  𝐸𝐷∗
+ (resp. , 𝐸𝐷∗

− ), i.e. E is obviously U.R.(resp., L.R.) by D∗ is QHC. Then by 

Lemma (3.1.) X is U.P.(resp., L.R.) 

(ii) ⇒ (i) From Theorem (4.1.). 

(i) ⇒ (iii) Let that (D×E, 𝔔) is 𝔽. 𝕎.U.T.S.(resp., 𝔽. 𝕎.L.T.S.) on (D,𝜌) in which 𝔔 = 𝜌 × 𝜏, then 

there is a projection X = π; (D×E, 𝔔) → (D,𝜌). We show that π is  closed and ∀d ∈ D, 𝐸𝐷
+(resp.,

𝐸𝐷
−) is U.R.(resp., L.R.)  in D×E. So, the result will be based on Theorem (3.1.). Let 𝒜 ⊂ D×E 

and a ∉ π(cl(𝒜)). ∀ e ∈E,(a,e) ∉ cl(𝒜), s𝑜 that ∃ a 𝜌−open 𝑎 𝜂ℙ𝕕 G of a 𝑎𝑛d a 𝜏-open a 𝜂ℙ𝕕 

𝔼e of e such that [𝔔 − cl(Ge × 𝐸𝑒
+(resp., 𝐸𝑒

−))] ∩ 𝒜 = ∅. Since E is QHC,{a}×E is a 𝔼.set in D 

×E. So that ∃ finitely many elements e1,e2,e3,...,en with, {a}×E⊂∪𝑘=1
𝑛 𝔔 − 𝑐𝑙(𝐺𝑒𝑘 × 𝐸𝑒𝑘

+ (resp.,

𝐸𝑒𝑘
− )). Currently, a ∈ ∩nk=1Ghk = G, which is a 𝜌 -open a 𝜂ℙ𝕕 of a ∫.t.(𝜌 −cl(G)∩π(𝒜) = ∅. So 

a ∉ clπ(𝒜) and thus clπ(𝒜) ⊂ π(cl(𝒜)). So π is closed by Lemma 

(2.1.). Next, let d ∈ D T.P. (𝐷 × 𝐸)𝑑
+(resp. , (𝐷 × 𝐸)𝑑

−)  = π−1(d) to be U.R.(resp., L.R.) in D 

×E. Let ℑ be a . on D ×E such that π−1(d) ∩ ad ℑ = ∅. ∀e ∈ E,(d,e) ∉ ad ℑ. So, ∃𝜌−open 

𝑎 𝜂ℙ𝕕 𝔘e of d in D, a 𝜌 -open 𝑎 𝜂ℙ𝕕 Ve of e in E and an 𝔽𝑒 ∈ ℑ such that F− cl(𝔘e ×Ve) ∩𝔽𝑒  

= ∅. As prove above, ∃finitely many elements e1,e2,e3,...,en of E such that {d}× 𝐸 ⊂∪𝑘=1
𝑛 𝔔 −

𝑐𝑙(𝐺𝑒𝑘 × 𝑉𝑒𝑘).  Putting 𝔘 and choosing 𝔽 ∈ ℑ with, 𝔽 ∩𝑘=1
𝑛  𝔽 𝑒𝑘, we get d ×E⊂ 𝔘 ×E⊂Q such 

that Q−cl(𝔘 ×E)∩ 𝔽 = ∅. Thus cl(𝔽)∩π−1(d) = ∅. So π−1(d) is U.R.(resp., L.R.)in D ×E. 

(iii)⇒(i) Taking D∗ = D, we have that X = π : D∗ × D → D∗ is U.R.(resp., L.R.) Therefore by 

(Theorem (3.5.)), D∗ ×E is an 𝔼.set and hence is QHC. 

 

C𝒐𝒓𝒐llary 4.7. For a topological space (E,𝜏), the next are equivalent: 

i. H 𝑖s QHC. 

ii. A 𝔽. 𝕎.M. (E,𝜏) is P.T space 𝑤𝑖th constant projection on D∗ in which D∗ is a singleton 

with two equal topologies meaning the unique topology on D∗. 

iii. The 𝔽. 𝕎.. (B×H,Q) is M.P.T.S. on (D,𝜌), in which 𝔔 = 𝜌 × 𝜏. 

 

 

 



IHJPAS. 36 (4) 2023 

406 
 

5.  Conclusion  

The main purpose of the present work is to providethe starting point for some application of 

fibr𝑒𝑤𝑖𝑠𝑒 multi-p𝑒rf𝑒𝑐t t𝑜p𝑜l𝑜gical spa𝑐𝑒𝑠 structures in a falter base by using multi-topological 

spaces. Definitions of characterization theorems are used for multi-r𝑖g𝑖d, fibr𝑒𝑤𝑖𝑠𝑒 multi-𝑤𝑒akly 

cl𝑜𝑠𝑒d, 𝔼 𝑠𝑒t, fibr𝑒𝑤𝑖𝑠e alm𝑜𝑠t multi-p𝑒rf𝑒ct, multi*-𝑐𝑜ntinu𝑜us fibr𝑒𝑤𝑖𝑠𝑒 multi∗ -t𝑜p𝑜l𝑜gical 

spa𝑐𝑒𝑠. 

 

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