EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 4, Article Number 5960
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

On Nearly 𝛼–Boundedness in 𝐿–Topological Spaces

Najah A. Alsaedi

Department of Mathematics, Faculty of Applied Science, Umm Al-Qura University, Makkah
Al Mukarramah, Saudi Arabia

Abstract. In this paper, we introduce and study the concept of nearly 𝛼–boundedness on
arbitrary 𝐿–subsets in 𝐿–topological spaces, which depends on the notion of 𝛼–regular closed
remoted neighborhood system. Several characterizations of nearly 𝛼–boundedness in terms of
convergence theory of 𝛼–filters, 𝛼–molecular nets and 𝛼–ideals are obtained. We prove that the
concept is a good extension, productive, and topologically invariant.

2020 Mathematics Subject Classifications: 54A40

Key Words and Phrases: Nearly 𝛼–boundedness, 𝛼–regular closed remoted neighborhood,
𝐿–topological space, 𝛼–filter, 𝛼–molecular nets, 𝛼–ideals, nearly 𝑄𝛼–compact

1. Introduction
Boundedness, as a natural generalization of relative compactness, was considered by

several authors (see [1] and [2]). In 1949, Hu [3] introduced the notion of boundedness in
general topological spaces and studied the closure, interior, base, and relativization of
boundedness. In-depth analysis of boundedness and its various weaker forms was done
by Lamprinos in [1] and [4]. A subset 𝐴 of a space 𝑋 is said to be bounded if every open
cover of 𝑋 has a finite subfamily that covers 𝐴. The concept of a bounded set is useful
in investigating non-regular topological spaces, since bounded sets in regular spaces are
compact.

In 1968, Chang [5] presented the concept of fuzzy compact. Since then, it has been a
very important topic to define proper fuzzy compactness. Many authors have written on
this problem and various kinds of fuzzy compactness have been presented [6, 7]. In 1984,
Li [8] introduced the fuzzy 𝑄𝛼–compactness based upon the concept of 𝑄–neighborhoods.
In 1992, Wang [9] generalized the 𝑄𝛼–compactness to the 𝐿–fuzzy topological spaces.

In 1997, Georgiou and Papadopoulos [10] gave a characterization of fuzzy nearly
compactness by using the notion of fuzzy weakly 𝜃–upper limit of fuzzy nets. Also,
he studied new fuzzy compactness and fuzzy boundedness in fuzzy topological spaces.
Recently, Georgiou and Papadopoulos in [11, 12] extended the concept of a bounded
set to fuzzy topology; and introduced the notion of fuzzy boundedness using the fuzzy
compactness given by Chang [5], which is not a good extension of ordinary compactness;
the Tychonoff product theorem does not hold, and it contradicts some kinds of separation

DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.5960

Email addresses: dr-Najah2008@hotmail.com, nasadi@uqu.edu.sa (N. A. Alsaedi)

https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0)



N. A. Alsaedi / Eur. J. Pure Appl. Math, 18 (4) (2025), 5960 2 of 22

axioms. Hence, the notion of fuzzy boundedness in [10] is not a good extension of ordinary
bounded, and so it is unsatisfactory.

In 2003, Nouh [13] introduced the concept of 𝑁–boundedness on an arbitrary 𝐿–
subset in 𝐿–topological spaces, and he gave new characterizations and properties of
𝑁–boundedness in terms of the convergence theory of 𝛼–nets, 𝛼–filters, and 𝛼–ideals.
He proved that the concept of 𝑁–boundedness is a good extension, productive, and
topologically invariant.

Since there are not enough studies on the concept of boundedness in 𝐿–topological
spaces. So in 2023, Alsaedi [14] introduced the concept of nearly Ω–boundedness on
an arbitrary 𝐿–subset in 𝐿–topological spaces by using the notion of Ω–upper limit of
Ω–nets.

In this paper, we generalize the nearly Ω–boundedness to nearly 𝛼–boundedness,
where we will study this concept on arbitrary 𝐿–subsets in 𝐿–topological spaces along
the line of nearly 𝑄𝛼–compactness defined by Wang [9] and 𝛼–regular closed remoted
neighborhood due to Zhao [15]. Then we give new characterizations and properties
of nearly 𝛼–boundedness in terms of the convergence theory of constant 𝛼–filter, 𝛼–
molecular nets, and 𝛼–ideals. We prove that the notion is a good extension, productive,
and topologically invariant.

2. Preliminaries

Throughout this paper 𝐿 = ⟨𝐿, ≤,∧,∨,∗ ⟩ denotes a completely distributive complete
lattice with a smallest element 0 and a largest element 1 (0 ≠ 1) and with an order–
reversing involution on it. An 𝛼 ∈ 𝐿 is called a molecule of 𝐿 if 𝛼 ≠ 0 and 0 ≤ 𝑣 ∨ 𝛾 ≤ 𝛼

implies 𝑣 ≤ 𝛾 or 𝛾 ≤ 𝑣, for all 𝑣, 𝛾 ∈ 𝐿. The set of all molecules of 𝐿 is denoted by 𝑀 (𝐿).
Let 𝑋 be a nonempty set. 𝐿𝑋 denotes the family of all mappings from 𝑋 to 𝐿. The

elements of 𝐿𝑋 are called 𝐿–subsets on 𝑋. 𝐿𝑋 can be made into a lattice by inducing the
order and involution from 𝐿. We denote the smallest element and the largest element
of 𝐿𝑋 by 0𝑋 and 1𝑋, respectively. If 𝛼 ∈ 𝐿, then the constant mapping 𝛼𝑋 : 𝑋 → {𝛼} is
𝐿–subset [16].

An 𝐿–point (or molecule on 𝐿𝑋), denoted by 𝑥𝛼, 𝛼 ∈ 𝑀 (𝐿) is a 𝐿–subset which is
defined by

𝑥𝛼 (𝑦) =
{
𝛼 : 𝑥 = 𝑦

0 : 𝑥 ≠ 𝑦

The family of all molecules of 𝐿𝑋 is denoted by 𝑀 (𝐿𝑋) [17]. For 𝜇 ∈ 𝐿𝑋 and
𝛼 ∈ 𝐿 we defined the set 𝜇𝑤𝛼 = {𝑥 ∈ 𝑋 : 𝜇(𝑥) ≥ 𝛼}, which it is called weak 𝛼–
cut of 𝜇. The set 𝜇𝑠𝛼 = {𝑥 ∈ 𝑋 : 𝜇(𝑥) ≰ 𝛼}, it is called strong 𝛼–cut of 𝜇 and
Supp(𝜇) = {𝑥 ∈ 𝑋 : 𝜇(𝑥) > 0} is called support of 𝜇 [18]. For any 𝜆 ∈ 𝐿𝑋 and
𝛼 ∈ 𝑀 (𝐿) with 𝛼′ ≥ 𝛼, we have (𝜆𝑤𝛼)′ ⊆ (𝜆′)𝑤𝛼. For Ψ ⊆ 𝐿𝑋, we define 2(Ψ) by the set
{𝜑 ⊆ Ψ : 𝜑 is finite subfamily of Ψ}.

An 𝐿–topology on 𝑋 is a subfamily 𝜏 of 𝐿𝑋 closed under arbitrary unions and finite
intersections. The pair (𝑋, 𝜏) is called an 𝐿–topological space (or 𝐿–ts, for short) [19].

If (𝐿𝑋, 𝜏) is an 𝐿–ts, then for 𝜂 ∈ 𝐿𝑋, 𝑐𝑙 (𝜂), 𝑖𝑛𝑡 (𝜂) and 𝜂′ will denote the closure,



N. A. Alsaedi / Eur. J. Pure Appl. Math, 18 (4) (2025), 5960 3 of 22

interior, and complement of 𝜂. A mapping 𝑓 : 𝐿𝑋 → 𝐿𝑌 is an 𝐿–valued Zadeh function
induced by a mapping 𝑓 : 𝑋 → 𝑌 , iff 𝑓 (𝜇) (𝑦) = ∨{𝜇(𝑥) : 𝑓 (𝑥) = 𝑦} for every 𝜇 ∈ 𝐿𝑋 and
every 𝑦 ∈ 𝑌 [17].

An 𝐿–ts (𝐿𝑋, 𝜏) is called fully stratified if for each 𝛼 ∈ 𝐿, 𝛼 ∈𝜏[18]. If (𝐿𝑋, 𝜏) is an
𝐿–ts, then the family of all crisp open sets in 𝜏 is denoted by [𝜏] i.e., (𝑋, [𝜏]) is a crisp
topological space [20].

Definition 2.1 [21]. If (𝐿𝑋, 𝜏) is 𝐿–ts, then 𝜇 ∈ 𝐿𝑋 is called a regular open set
iff 𝜇 = 𝑖𝑛𝑡 (𝑐𝑙 (𝜇)). The family of all regular open sets is denoted by 𝑅𝑂 (𝐿𝑋, 𝜏). The
complement of a regular open set is called a regular closed set and satisfies 𝜇𝑐 = 𝑐𝑙 (𝑖𝑛𝑡 (𝜇)).
The family of all regular closed sets is denoted by 𝑅𝐶(L𝑋, 𝜏).

Definition 2.2 [21]. The 𝐿–valued Zadeh mapping 𝑓𝐿 : (𝐿𝑋, 𝜏) → (𝐿𝑌 ,Δ) is called
Almost 𝐿–continuous iff 𝑓 −1

𝐿
(𝜂) ∈ 𝜏′ for each 𝜂 ∈ 𝑅𝐶 (𝐿𝑌 ,Δ).

Definition 2.3 [9]. Let (𝐿𝑋, 𝜏) be an 𝐿–ts and 𝑥𝛼 ∈ 𝑀 (𝐿𝑋). Then 𝜆 ∈ 𝜏′ is called
an remoted neighborhood (R–nbd, for short) of 𝑥𝛼 if 𝑥𝛼 ∉ 𝜆. The set of all R–nbds of 𝑥𝛼
is called remoted neighborhood system and is denoted by 𝑅𝑥𝛼 .

Definition 2.4 [15]: Let (𝐿𝑋, 𝜏) be an 𝐿-ts, 𝜇 ∈ 𝐿𝑋 and 𝛼 ∈ 𝑀 (𝐿). Ψ ⊂ 𝜏′ is called
an:

(i) 𝛼-remoted neighborhood family of 𝜇, briefly 𝛼-RF of 𝜇, if for each 𝐿–point 𝑥𝛼 ∈ 𝜇

there is 𝜆 ∈ Ψ such that 𝜆 ∈ 𝑅𝑥𝛼 .
(ii) �̄�-remoted neighborhood family of 𝜇, briefly �̄�-RF of 𝜇, if there exists 𝛾 ∈ 𝛽∗(𝛼)

such that Ψ is a 𝛾-RF of 𝜇, where 𝛽∗(𝛼) = 𝛽(𝛼) ∩ 𝑀 (𝐿), and 𝛽(𝛼) denotes the union of
all the minimal sets relative to 𝛼.

Definition 2.5 [22]: Let (𝐿𝑋, 𝜏) be an 𝐿-ts, 𝜇 ∈ 𝐿𝑋 and 𝛼 ∈ 𝑀 (𝐿). Then Ψ ⊂
𝑅𝐶 (𝐿𝑋, 𝜏) is called an 𝛼-regular closed remoted neighborhood family of 𝜇, briefly 𝛼–RCRF
of 𝜇, if for each 𝐿-point 𝑥𝛼 ∈ 𝜇 there is 𝜆 ∈ Ψ such that 𝜆 ∈ 𝑅𝑥𝛼 .

Definition 2.6 [21]. Let (𝐿𝑋, 𝜏) be an 𝐿-ts, 𝜇 ∈ 𝐿𝑋 and 𝛼′ ∈ 𝑀 (𝐿). Then the family
Ψ ⊆ 𝜏 is called an:

(i) 𝛼–cover of 𝜇, if for each 𝑥 ∈ 𝜇𝑤𝛼′ there is 𝜆 ∈ Ψ such that 𝜆(𝑥) ≰ 𝛼.
(ii) Nearly 𝛼–cover of 𝜇, if for each 𝑥 ∈ 𝜇𝑤𝛼′ there is 𝜆 ∈ Ψ such that int(cl(𝜆)) (𝑥) ≰ 𝛼.

Definition 2.7 [17]. Let (𝐿𝑋, 𝜏) be an 𝐿-ts and 𝜇 ∈ 𝐿𝑋. Then 𝑥𝛼 ∈ 𝑀 (𝐿𝑋) is called
the 𝛿–adherent point of 𝜇 and write 𝑥𝛼 ∈ 𝛿𝑐𝑙 (𝜇) iff 𝜇 ≰ 𝑐𝑙 (int(𝜆)) for each 𝜆 ∈ 𝑅𝑥𝛼 . If
𝜇 = 𝛿𝑐𝑙 (𝜇), then 𝜇 is called a 𝛿–closed 𝐿–subset. The family of all 𝛿–closed 𝐿–subsets of
𝑋 is denoted by 𝛿𝐶 (𝐿𝑋, 𝜏) and its complement is called the family of all 𝛿–open 𝐿–subsets
and denoted by 𝛿𝑂 (𝐿𝑋, 𝜏).

Definition 2.8 [23]. Let (𝐿𝑋, 𝜏) be an 𝐿-ts, 𝜇 ∈ 𝐿𝑋 and 𝛼 ∈ 𝑀 (𝐿). An 𝛼-RF
Ψ = {𝜂 𝑗 : 𝑗 ∈ 𝐽} of 𝜇 is called a directed if 𝜂1, 𝜂2 ∈ Ψ there is 𝜂3 ∈ Ψ such that
𝜂3 ≤ 𝜂1 ∧ 𝜂2.

Definition 2.9 [9]: Let (𝐷, ≤) be a directed set. Then the mapping 𝑆 : 𝐷 → 𝐿𝑋 and
denoted by 𝑆 = {𝜇𝑛 : 𝑛 ∈ 𝐷} is called a net of 𝐿-subsets in 𝑋. Specifically, the mapping



N. A. Alsaedi / Eur. J. Pure Appl. Math, 18 (4) (2025), 5960 4 of 22

𝑆 : 𝐷 → 𝑀 (𝐿𝑋) is said to be a molecular net in 𝐿𝑋. If 𝜇 ∈ 𝐿𝑋 and for each 𝑛 ∈ 𝐷, 𝑆 ∈ 𝜇

then 𝑆 is called a net in 𝜇.

Definition 2.10 [9]: Let (𝐿𝑋, 𝜏) be an 𝐿-ts and 𝑆 = {𝑆(𝑛) : 𝑛 ∈ 𝐷} be a molecular
net in 𝐿𝑋. 𝑆 is called a molecular 𝛼-net (𝛼 ∈ 𝑀 (𝐿)), if for each 𝛾 ∈ 𝛽∗(𝛼) there exists
𝑛 ∈ 𝐷 such that ∨(𝑆(𝑚)) ≥ 𝛾 whenever 𝑚 ≥ 𝑛, where ∨(𝑆(𝑚)) is the height of the
molecular 𝑆(𝑚). If ∨(𝑆(𝑚)) = 𝛼 for each 𝑚 ∈ 𝐷, then {𝑆(𝑚) : 𝑚 ∈ 𝐷} is called a constant
molecular 𝛼-net.

Definition 2.11 [9]: Let 𝑆 = {𝑆(𝑛) : 𝑛 ∈ 𝐷} and 𝑇 = {𝑇 (𝑚) : 𝑚 ∈ 𝐸} be molecular
nets in (𝐿𝑋, 𝜏). Then 𝑇 is said to be a molecular subnet of 𝑆 if there is a mapping
𝑓 : 𝐸 → 𝐷 that satisfies the following conditions:
(i) 𝑇 = 𝑆 ◦ 𝑓

(ii) For each 𝑛 ∈ 𝐷 there is 𝑚 ∈ 𝐸 such that 𝑓 (𝑙) ≥ 𝑛 for each 𝑙 ∈ 𝐸 , 𝑙 ≥ 𝑚.

Definition 2.12 [21]: Let (𝐿𝑋, 𝜏) be an 𝐿–ts and Δ = {𝜇𝑛 : 𝑛 ∈ 𝐷} be a net of
𝐿–subsets in (𝐿𝑋, 𝜏) and 𝑥𝛼 ∈ 𝑀 (𝐿𝑋).

Then:
(i) 𝑥𝛼 is called the 𝛿–limit point of Δ (or 𝛿–converges) to the point 𝑥𝛼, in symbols

Δ
𝛿−−→ 𝑥𝛼 if for every 𝜂 ∈ 𝑅𝑥𝛼 there is an 𝑛 ∈ 𝐷 such that for every 𝑚 ∈ 𝐷 and 𝑚 ≥ 𝑛 then

𝜇𝑛 ∉ 𝑐𝑙 (int(𝜂)). The union of all 𝛿–limit points of Δ are denoted by 𝛿.lim(Δ).

(ii) 𝑥𝛼 is called a 𝛿–cluster (𝛿–adherent) point of Δ, in symbols Δ
𝛿∝ 𝑥𝛼 if for every 𝜂 ∈ 𝑅𝑥𝛼

and every 𝑛 ∈ 𝐷 there exists 𝑚 ∈ 𝐷 such that 𝑚 ≥ 𝑛 and 𝜇𝑛 ∉ 𝑐𝑙 (int(𝜂)). The union of
all 𝛿–cluster points of Δ are denoted by 𝛿.lim(Δ).

If 𝛿.lim(Δ)=𝛿.lim(Δ) = 𝜇, then we say that 𝜇 is the 𝛿–limit of Δ, or we say that Δ

𝛿–converges to 𝜇, in symbol 𝛿.lim(Δ) = 𝜇.
The 𝛿–limit and 𝛿–cluster points of a molecular net are defined similarly in [16].

Definition 2.13 [17]: Let (𝐿𝑋, 𝜏) be an 𝐿-ts and 𝑆 be a molecular 𝛼-net in 𝐿𝑋.
Then 𝑥𝛼 ∈ 𝑀 (𝐿𝑋) is called the 𝛿–limit point of 𝑆, (or 𝑆 𝛿–converges to 𝑥𝛼) in symbol

𝑆
𝛿−→ 𝑥𝛼 if for every 𝜇 ∈ 𝑅𝑥𝛼 there is an 𝑛 ∈ 𝐷 such that for each 𝑚 ∈ 𝐷 and 𝑚 ≥ 𝑛 we

have 𝑆(𝑚) ≰ 𝑐𝑙 (int(𝜇)). The union of all limit points of 𝑆 is denoted by 𝛿.lim(𝑆).
Definition 2.14 [17]: Let (𝐿𝑋, 𝜏) be an 𝐿–ts and 𝑆 be a molecular 𝛼–net in 𝐿𝑋.

Then 𝑥𝛼 ∈ 𝑀 (𝐿𝑋) is called a 𝛿–cluster point of 𝑆, in symbol 𝑆
𝛿∝ 𝑥𝛼 if for every 𝜇 ∈ 𝑅𝑥𝛼

and every 𝑛 ∈ 𝐷 there is 𝑚 ∈ 𝐷 such that 𝑚 ≥ 𝑛 and 𝑆(𝑚) ∉ 𝑐𝑙 (int(𝜇)). The union of all
𝛿–cluster points of 𝑆 is denoted by 𝛿𝑎𝑑ℎ(𝑆).

Theorem 2.15 [22]: Assume that 𝑆 = {𝑆(𝑛) : 𝑛 ∈ 𝐷} is a molecular net in an 𝐿–ts
(𝐿𝑋, 𝜏) and 𝑥𝛼 ∈ 𝑀 (𝐿𝑋). Then the following results are true:

(i) 𝑆
𝛿∝ 𝑥𝛼 iff there exists a subnet 𝑇 of 𝑆 such that 𝑇

𝛿−−→ 𝑥𝛼.

(ii) If 𝑆
𝛿−−→ 𝑥𝛼, then 𝑇

𝛿−−→ 𝑥𝛼 for each subnet 𝑇 of 𝑆.

Definition 2.16 [8]: Let (𝐿𝑋, 𝜏) be an 𝐿-ts, 𝜇 ∈ 𝐿𝑋. Then 𝜇 is called nearly
𝑄𝛼–compact (or 𝑁𝑄𝛼–compact) in (𝐿𝑋, 𝜏) if for each 𝛼 ∈ 𝑀 (𝐿) and every 𝛼–RF Ψ of 𝜇
there is Ψ𝑜 ∈ 2(Ψ) such that Ψ𝑜 is an 𝛼–RCRF of 𝜇.



N. A. Alsaedi / Eur. J. Pure Appl. Math, 18 (4) (2025), 5960 5 of 22

If 1𝑋 is nearly 𝑄𝛼–compact, then (𝐿𝑋, 𝜏) is called a nearly 𝑄𝛼–compact space.

Definition 2.17 [23]: An 𝐿–ts (𝐿𝑋, 𝜏) is said to be:
(i) 𝐿𝑇1–space iff for any 𝑥𝛼, 𝑦𝛾 ∈ 𝑀 (𝐿𝑋), 𝑥 ≠ 𝑦 there is 𝜆 ∈ 𝑅𝑥𝛼 such that 𝑦𝛾 ∈ 𝜆.
(ii) 𝐿𝑇2–space iff for any 𝑥𝛼, 𝑦𝛾 ∈ 𝑀 (𝐿𝑋), 𝑥 ≠ 𝑦 there is 𝜆 ∈ 𝑅𝑥𝛼 , 𝜂 ∈ 𝑅𝑦𝛾 such that
𝜆 ∨ 𝜂 = 1𝑋.
(iii) 𝐿𝑇2 1

2
–space iff for any 𝑥𝛼, 𝑦𝛾 ∈ 𝑀 (𝐿𝑋), 𝑥 ≠ 𝑦 there is 𝜆 ∈ 𝑅𝑥𝛼 , 𝜂 ∈ 𝑅𝑦𝛾 such that

int(𝜆) ∨ int(𝜂) = 1𝑋.
(iv) 𝐿𝑅2–space (regular space) iff for all 𝛼 ∈ 𝑀 (𝐿), 𝑥 ∈ 𝑋 and for each 𝜆 ∈ 𝑅𝑥𝛼 there is
𝜂 ∈ 𝑅𝑥𝛼 , 𝜌 ∈ 𝜏′ such that 𝜂 ∨ 𝜌 = 1𝑋 and 𝜆 ∧ 𝜌 = 0𝑋.
(v) 𝐿𝑆𝑅2–space (Semi-regular space) iff for all 𝑥𝛼 ∈ 𝑀 (𝐿𝑋) and for each 𝜆 ∈ 𝑅𝑥𝛼 there is
𝜂 ∈ 𝑅𝑥𝛼 such that 𝜆 ≤ 𝑐𝑙 (int(𝜂)).
(vi) 𝐿𝑇3–space iff it is 𝐿𝑅2–space and 𝐿𝑇1–space.

Theorem 2.18 [8]: Let (𝐿𝑋, 𝜏) be an 𝐿–ts and 𝜇 ∈ 𝐿𝑋. Then the following properties
are true:

(i) Every set with finite support is nearly 𝑄𝛼–compact.
(ii) Every nearly 𝑄𝛼–compact set in a fully stratified and 𝐿𝑇2–space , then it is

𝛿–closed.

Theorem 2.19 [8]: Let (𝐿𝑋, 𝜏) be an 𝐿–ts, 𝛼 ∈ 𝑀 (𝐿) and 𝜇 ∈ 𝐿𝑋. Then 𝜇 is
𝑁𝑄𝛼–compact iff for each constant molecular 𝛼–net 𝑆 contained in 𝜇 has a 𝛿–cluster
point with height 𝛼 in 𝜇.

Theorem 2.20 [21]. If 𝐿–ts (𝐿𝑋, 𝜏) is 𝐿𝑅2–space, then it is 𝐿𝑆𝑅2–space.

Theorem 2.21 [21]: An 𝐿–ts (𝐿𝑋, 𝜏) is 𝐿𝑆𝑅2–space iff for any 𝜇 ∈ 𝐿𝑋, 𝑐𝑙 (𝜇) = 𝛿𝑐𝑙 (𝜇).

Corollary 2.22 [19]. If 𝐿–ts (𝐿𝑋, 𝜏) is 𝐿𝑆𝑅2–space, then a closed 𝐿–subset is a
𝛿–closed 𝐿–subset and hence 𝛿𝑐𝑙 (𝜇) is a 𝛿–closed 𝐿–subset.

Definition 2.23 [15]: The nonempty family F ⊂ 𝐿𝑋 is called an 𝐿–filter if the
following conditions are satisfied, for each 𝜇1, 𝜇2 ∈ 𝐿𝑋

(i) 0𝑋 ∉ F
(ii) If 𝜇1 ≤ 𝜇2 and 𝜇1 ∈ F , then 𝜇2 ∈ F .
(iii) If 𝜇1, 𝜇2 ∈ F , then 𝜇1 ∧ 𝜇2 ∈ F .

Definition 2.24 [15]: A filter F in 𝐿𝑋 is called an 𝛼–filter (𝛼 ∈ 𝑀 (𝐿)), if for every
𝜆 ∈ F ,

∨
𝑥∈𝑋 𝜆(𝑥) ≥ 𝛼.

Definition 2.25 [15]: Let (𝐿𝑋, 𝜏) be an 𝐿–ts and F be an 𝐿–filter in 𝐿𝑋.

Then 𝑥𝛼 ∈ 𝑀 (𝐿𝑋) is called the 𝛿–cluster point of F , in symbol F 𝛿∝ 𝑥𝛼 if for each
𝜆 ∈ F and each 𝜇 ∈ 𝑅𝑥𝛼 , 𝜆 ⊈ 𝑐𝑙 (int(𝜇)). The union of all 𝛿–cluster points of F is denoted
by 𝛿𝑎𝑑ℎ(F ).

Definition 2.26 [27]: The nonempty family 𝐼 ⊂ 𝐿𝑋 is called an 𝐿–ideal if the
following conditions are satisfied, for each 𝜇1, 𝜇2 ∈ 𝐿𝑋

(i) 1𝑋 ∉ 𝐼

(ii) If 𝜇1 ≤ 𝜇2 and 𝜇2 ∈ 𝐼, then 𝜇1 ∈ 𝐼.



N. A. Alsaedi / Eur. J. Pure Appl. Math, 18 (4) (2025), 5960 6 of 22

(iii) If 𝜇1, 𝜇2 ∈ 𝐼, then 𝜇1 ∨ 𝜇2 ∈ 𝐼.

Definition 2.27 [27]: Let 𝐼 be an 𝐿–ideal in an 𝐿–ts (𝐿𝑋, 𝜏) and 𝛼 ∈ 𝑀 (𝐿). Then 𝐼

is said to be an 𝛼–ideal, if ∨𝑛∈𝑋𝜂(𝑥) < 𝛼 for each 𝜂 ∈ 𝐼.

Theorem 2.28 [22]: Let F be a 𝐿–filter in an 𝐿–ts (𝐿𝑋, 𝜏) and 𝑆(F ) be the
𝐿–molecular net induced by F . Then 𝛿𝑎𝑑ℎ(F ) = 𝛿𝑎𝑑ℎ(𝑆(F )).

Theorem 2.29 [22]: Suppose that 𝑆 is a 𝐿–net in an 𝐿–ts (𝐿𝑋, 𝜏) and F (𝑆) is the
𝐿–filter induced by 𝑆. Then 𝛿𝑎𝑑ℎ(𝑆) = 𝛿𝑎𝑑ℎ(F (𝑆)).

Theorem 2.30 [22]: Suppose that 𝐼 is an 𝐿–ideal in an 𝐿–ts (𝐿𝑋, 𝜏), and 𝑆(𝐼) is
the 𝐿–molecular net induced by 𝐼. Then 𝛿𝑎𝑑ℎ(𝐼) = 𝛿𝑎𝑑ℎ(𝑆(𝐼)).

3. Nearly 𝛼–Boundedness in 𝐿–topological spaces
In this section, we introduce the concept of nearly 𝛼–bounded sets in 𝐿–topological

spaces. Then we obtain several characterizations of nearly 𝛼–bounded sets.
Definition 3.1. Let (𝐿𝑋, 𝜏) be an 𝐿−ts, 𝜇 ∈ 𝐿𝑋 and 𝛼 ∈ 𝑀 (𝐿), then 𝜇 ∈ 𝐿𝑋 is called

a nearly 𝛼−bounded (𝑁𝛼−bounded, for short) set in (𝐿𝑋, 𝜏) iff for each 𝛼 − 𝑅𝐹 Ψ of 1𝑋,
there exists Ψ𝑜 ∈ 2(Ψ) such that Ψ𝑜 is an 𝛼 − 𝑅𝐶𝑅𝐹 of 𝜇.

Theorem 3.2. Suppose that 𝑓𝐿 : (𝐿𝑋, 𝜏) → (𝐿𝑌 ,Δ) is a 𝐿−continuous and 𝜇 ∈ 𝐿𝑋 is
a 𝑁𝛼−bounded 𝐿−subset in (𝐿𝑋, 𝜏), then 𝑓𝐿 (𝜇) is a 𝑁𝛼−bounded 𝐿−subset in (𝐿𝑌 ,Δ).

Proof. Let 𝜇 be a 𝑁.𝛼–bounded in 𝐿𝑋 and let Ψ ⊂ Δ′ be an 𝛼–RF of 1𝑌 (𝛼 ∈ 𝑀 (𝐿)).
To begin with, let us show that 𝑓 −1

𝐿
(Ψ) = { 𝑓 −1

𝐿
(𝜆) : 𝜆 ∈ Ψ} is an 𝛼–RF of 1𝑋. Since 𝑓𝐿

is a 𝐿–continuous, then 𝑓 −1
𝐿

(Ψ) ⊂ 𝜏′. Let 𝑥 ∈ 𝑋, then 𝑓𝐿 (𝑥𝛼) = ( 𝑓 (𝑥))𝛼 ∈ 𝑓𝐿 (1𝑋) and
by Ψ ⊂ Δ′ is an 𝛼–RF of 1𝑌 there exists 𝜆 ∈ Ψ with 𝜆 ∈ 𝑅( 𝑓 (𝑥 ) )𝛼 , i.e, ( 𝑓 (𝑥))𝛼 ∉ 𝜆 or,
equivalently, 𝜆( 𝑓 (𝑥)) ≱ 𝛼. By the definition of inverse mapping, 𝑓 −1

𝐿
(𝜆) (𝑥) = 𝜆( 𝑓 (𝑥)) ≱ 𝛼,

hence 𝑥𝛼 ∉ 𝑓 −1
𝐿

(𝜆). It follows that 𝑓 −1
𝐿

(𝜆) ∈ 𝑅𝑥𝛼 . Therefore 𝑓 −1
𝐿

(Ψ) is an 𝛼–RF of
1𝑋. From the 𝑁.𝛼–boundedness of 𝜇 there exists Ψ◦ ∈ 2(Ψ) such that 𝑓 −1

𝐿
(Ψ◦) is an

𝛼–RCRF of 𝜇, that is, for each 𝑥𝛼 ∈ 𝜇 there exists 𝜆 ∈ Ψ such that 𝑓 −1
𝐿

(𝜆) ∈ 𝑅𝑥𝛼 , i.e.,
𝑓 −1
𝐿

(𝜆) (𝑥) ≱ 𝛼. Hence 𝜆(𝑦) = 𝜆( 𝑓 (𝑥)) ≱ 𝛼 and so for each 𝑦𝛼 ∈ 𝑓𝐿 (𝜇), there exists 𝑥𝛼 ∈ 𝜇

and 𝜆 ∈ Ψ◦ satisfying 𝑦𝛼 = 𝑓𝐿 (𝑥𝛼) ∉ 𝜆. Hence 𝜆(𝑦) ≱ 𝛼, i.e., 𝜆 ∈ 𝑅𝑦𝛼 . This implies that
Ψ◦ ∈ 2(Ψ) is an 𝛼–RCRF of 𝑓𝐿 (𝜇). By Definition 3.1, we have 𝑓𝐿 (𝜇) is a 𝑁.𝛼–bounded
𝐿–subset in (𝐿𝑌 ,Δ).

Theorem 3.3. Let (𝐿𝑋, 𝜏) be an 𝐿–ts and 𝛼′ ∈ 𝑀 (𝐿), then the set 𝜇 ∈ 𝐿𝑋 is
𝑁𝛼–bounded iff for every 𝛼–cover Ψ ⊆ 𝜏 of 1𝑋 there exists Ψ𝑜 ∈ 2(Ψ) such that Ψ𝑜 is a
nearly 𝛼–cover of 𝜇.

Proof. Let 𝜇 ∈ 𝐿𝑋 be a 𝑁.𝛼–bounded set and let Ψ ⊆ 𝜏 is any 𝛼–cover of 1𝑋. Let
𝜑 = Ψ′ = {𝜆′ : 𝜆 ∈ Ψ} and let 𝛾 = 𝛼′. One can see that 𝜑 is an 𝛾–RF of 1𝑋. Since
1𝑋 (𝑥) ≥ 𝛾 for each 𝑥𝛾 ∈ 1𝑋, i.e., 𝑥 ∈ 𝑋 for each 𝑥𝛾 ∈ 1𝑋, there exists 𝜆 ∈ Ψ satisfying
𝜆(𝑥) ≱ 𝛼 = 𝛾′, this equivalently, there exists 𝜆′ ∈ 𝜑 with 𝛾 ≰ 𝜆′(𝑥), and so 𝜆′ ∈ 𝑅𝑥𝛾 . This

implies that 𝜑 is an 𝛾–RF of 1𝑋. Being 𝜇 is 𝑁.𝛼–bounded, then there exists 𝜑◦ ∈ 2(𝜑)

such that 𝜑◦ is an 𝛾–RCRF of 𝜇. We assert 𝜑′
◦ ∈ 2(Ψ) is a nearly 𝛼–cover of 𝜇. In fact,

for each 𝑥𝛾 ∈ 𝜇 there is 𝜆′ ∈ 𝜑◦ satisfying 𝑐𝑙 (int(𝜆′)) ∈ 𝑅𝑥𝛾 , that is 𝛾 ≰ 𝑐𝑙 (int(𝜆′(𝑥))),
equivalently, for each 𝑥 ∈ 𝜇𝑤𝛾

we have 𝜆 ∈ 𝜑′
◦ with int(𝑐𝑙 (𝜆(𝑥))) = (𝑐𝑙 (int(𝜆′)) (𝑥))′ ≰ 𝛼.

Therefore, 𝜑′
◦ is a nearly 𝛼–cover of 𝜇.



N. A. Alsaedi / Eur. J. Pure Appl. Math, 18 (4) (2025), 5960 7 of 22

Conversely, suppose that the condition is satisfied and let that Ψ is an 𝛾–RF of 1𝑋.
Put Ψ′ = 𝜑 and 𝛾′ = 𝛼, then 𝜑 is an 𝛼–cover of 1𝑋, and then there exists 𝜑◦ ∈ 2(𝜑) such
that 𝜑◦ is a nearly 𝛼–cover of 𝜇. Evidently, 𝜑′

◦ ∈ 2(Ψ) is an 𝛾–RCRF of 𝜇. Hence 𝜇 is a
𝑁.𝛼–bounded.

Theorem 3.4. Let (𝐿𝑋, 𝜏) be an 𝐿–ts, 𝜇 ∈ 𝐿𝑋 is a 𝛼–bounded [23], then 𝜇 is a
𝑁𝛼–bounded.

proof It follows directly from the fact that 𝑅𝐶 (𝐿𝑋, 𝜏) ⊆ 𝜏′.
The following example shows that the converse is not true in general.
Example 3.5. Let 𝐿 = [0, 1], 𝑋 = 𝑁 and let 𝜏 = {0𝑋, 𝑥.5, 1𝑋}. Then (𝐿𝑋, 𝜏) is

𝐿–ts. Firstly, we show that 1𝑋 is not 𝛼–bounded set. In fact, we suppose that constant
0.5–net 𝑆 = {𝑥.5 : 𝑥 ∈ 𝑋} in 1𝑋, Let 𝑦0.3 ∈ 𝑀 (𝐿𝑋), 𝑥 ≠ 𝑦, then 𝑅𝑦0.3 = {0𝑋, 𝑥0.5}. Since
𝑆(𝑛) = 𝑥0.5 ≤ 𝑥0.5 ∈ 𝑅𝑦0.3 . So 𝑦0.3 is not cluster point of 𝑆 in 1𝑋. On account of the
arbitrariness of 𝑦 it follows that the 𝑆 has no cluster point in 1𝑋 with height 0.3. Thus 𝜇

is not 𝛼–bounded set.

Now, we show that 1𝑋 is 𝑁𝛼–bounded set. Let 𝑆 = {𝑥𝛼 : 𝑥 ∈ 𝑋, 𝛼 ∈ 𝐿} is any constant
𝛼–net in 1𝑋.

(i) If 𝛼 > 0.5 then 𝑅𝑥𝛼 = {0𝑋, 𝑥0.5}, where 𝑐𝑙 (𝑖𝑛𝑡 (𝑥0.5)) = 𝑥0.5. Since 𝑆(𝑛) = 𝑥𝛼 ∉

0𝑋 ∀ 𝑛 ∈ 𝑁, ∀𝛼 > 0.5. Hence, 𝑥𝛼 is 𝛿–cluster point of 𝑆 with height 𝛼 in 1𝑋.

(ii) If 𝛼 ≤ 0.5, then 𝑅𝑥𝛼 = {0𝑋} where 𝑐𝑙 (𝑖𝑛𝑡 (0𝑋)) = 0𝑋. Since 𝑆(𝑛) = 𝑥𝛼 ∉ 0𝑋 ∀ 𝑛 ∈
𝑁, ∀𝛼 ≤ 0.5. So 𝑥𝛼 is 𝛿–cluster point of 𝑆 with height 𝛼 in 1𝑋. Thus 1𝑋 is
𝑁𝛼–bounded set.

Theorem 3.6. (The goodness of 𝛼−boundedness) Let (𝐿𝑋𝑖𝜔𝐿 (𝑇)) be the induced
𝐿-ts by the ordinary space (𝑋,𝑇), 𝛼 ∈ 𝑀 (𝐿) and 𝜇 ∈ 𝐿𝑋. Then 𝜇 is 𝑁𝛼−bounded in
(𝐿𝑋𝑖 , 𝜔𝐿 (𝑇)) iff 𝜇𝜔𝛼 = {𝑥 ∈ 𝑋 : 𝜇(𝑥) ≥ 𝛼} is nearly bounded in (𝑋,𝑇).

proof Let 𝜇 ∈ 𝐿𝑋 be a 𝑁𝛼−bounded and {𝑈 𝑗 : 𝑗 ∈ 𝐽} be an open cover of 𝑋 in
(𝑋,𝑇). Then the family {1𝑈 𝑗

: 𝑗 ∈ 𝐽} is a 𝛼−covr of 1𝑋 in (𝐿𝑋𝑖 , 𝜔𝐿 (𝑇)). Since 𝜇 is
𝑁𝛼−bounded, there is a finite subset 𝐽𝑜 of 𝐽 such that {1𝑈 𝑗

: 𝑗 ∈ 𝐽𝑜} is an nearly 𝛼−cover
of 𝜇 in (𝐿𝑋𝑖𝜔𝐿 (𝑇)) in line with Theorem 3.3, i.e, for each 𝑥 ∈ 𝜇𝜔𝛼 there exists 𝑗 ≤ 𝑛

and 1𝑈 𝑗
∈ 𝜑𝑜 satisfying int(𝑐𝑙 (1𝑈 𝑗

)) (𝑥) ≰ 𝛼. However int(𝑐𝑙 (1𝑈 𝑗
)) (𝑥) = 1int(𝑐𝑙 (𝑈 𝑗 ) ) , and

so 𝑥 ∈ int(𝑐𝑙 (𝑈 𝑗)). This implies that
⋃𝑛

𝑗=1 int(𝑐𝑙 (𝑈 𝑗)) ⊃ 𝜇𝜔𝛼. Hence 𝜇𝜔𝛼 is a nearly
bounded in (𝑋,𝑇) for any 𝛼 ∈ 𝑀 (𝐿).

Conversely, suppose that 𝜇𝜔𝛼 is a nearly bounded set for any 𝛼 ∈ 𝑀 (𝐿) and 𝜑 is a
𝛼−cover of 1𝑋 in (𝐿𝑋𝑖𝜔𝐿 (𝑇)). Then for any 𝑥 ∈ 𝑋, 𝛼 ∈ 𝑀 (𝐿) there exists 𝜂𝑥 ∈ 𝜑 such
that 𝜂𝑥 (𝑥) ≰ 𝛼. Put (𝜂𝑥)𝑠𝛼 = {𝑦 ∈ 𝑋 : 𝜂𝑥 (𝑦) ≰ 𝛼}, then 𝑥 ∈ (𝜂𝑥)𝑠𝛼. Since 𝜂𝑥 ∈ 𝜔𝐿 (𝑇)
then (𝜂𝑥)𝑠𝛼 ∈ 𝑇 . One can see that 𝑈 = {(𝜂𝑥)𝑠𝛼 : 𝑥 ∈ 𝑋} is an open cover of 𝑋 in (𝑋,𝑇).
Since 𝜇𝜔𝛼 is a nearly bounded in (𝑋,𝑇), then there exists 𝑥1, 𝑥2, ..., 𝑥𝑛 ∈ 𝜇𝑠𝛼 such that
𝑈𝑜 = {(𝜂𝑥𝑖 )𝑠𝛼 : 𝑖 = 1, 2, ..., 𝑛} is an nearly open cover of 𝜇𝜔𝛼. Thus there exists 𝑖 ≤ 𝑛 with
𝑥 ∈ int(𝑐𝑙 ((𝜂𝑥𝑖 )𝑠𝛼)) for each 𝑥 ∈ 𝜇𝜔𝛼. However int(𝑐𝑙 ((𝜂𝑥𝑖 )𝑠𝛼)) ⊂ int(𝑐𝑙 ((𝜂𝑥𝑖 )))𝑠𝛼 and so
𝑥 ∈ int(𝑐𝑙 ((𝜂𝑥𝑖 )))𝑠𝛼, thus int(𝑐𝑙 (𝜂𝑥𝑖 )) (𝑥) ≰ 𝛼. Hence 𝜑𝑜 = {𝜂𝑥𝑖 : 𝑖 = 1, 2, ..., 𝑛} ∈ 2(𝜑) is a
nearly 𝛼−cover of 𝜇. According to Theorem 3.3, 𝜇 is 𝑁𝛼−bounded in (𝐿𝑋𝑖 , 𝜔𝐿 (𝑇)).



N. A. Alsaedi / Eur. J. Pure Appl. Math, 18 (4) (2025), 5960 8 of 22

Theorem 3.7. Let (𝐿𝑋, 𝜏) be a 𝐿−ts. and let 𝜇 ∈ 𝐿𝑋. If 𝜂 is 𝑁𝛼−bounded and
𝜇 ≤ 𝜂, then 𝜇 is a 𝑁𝛼−bounded.

Proof. Let 𝜂 be a 𝑁𝛼−bounded set and 𝜇 ≤ 𝜂. Let Ψ ⊂ 𝜏′ be an 𝛼−RF of 1𝑋. Since
𝜂 is 𝑁𝛼−bounded set, then there exists a finite subfamily Ψ𝑜 ∈ 2(Ψ) such that Ψ𝑜 is an
𝛼−RCRF of 𝜂, since 𝜇 ≤ 𝜂, then Ψ𝑜 is an 𝛼−RCRF of 𝜇 and so 𝜇 is a 𝑁𝛼−bounded set.

Definition 3.8. Let (𝐿𝑋, 𝜏) be an 𝐿−ts and 𝑥𝛼 ∈ 𝑀 (𝐿𝑋). If 𝜇 ∈ 𝐿𝑋 is closed and
𝑁𝛼−bounded set, then 𝜇 is called the 𝑁𝛼𝐵−remoted neighborhood of 𝑥𝛼 (𝑁𝛼𝐵𝑅−nbd,
for short) of 𝑥𝛼 if 𝑥𝛼 ∉ 𝜇. The set of all 𝑁𝛼𝐵𝑅−nbds of 𝑥𝛼 is denoted by 𝑁𝛼𝐵𝑅𝑥𝛼 .
We note that [24] 𝛼𝐵𝑅𝑥𝛼 ⊆ 𝑁𝛼𝐵𝑅𝑥𝛼 ⊆ 𝑅𝑥𝛼 , ∀ 𝑥𝛼 ∈ 𝑀 (𝐿𝑋).
The following example shows that the converse is not true in general.

Example 3.9 Let 𝐿 = [0, 1], 𝑋 = □ and let 𝜏 = {0𝑋, 𝑥.5, 1𝑋}. Then (𝐿𝑋, 𝜏) is 𝐿−ts.
Firstly, we show that 𝜇 = 𝑥.4 ∈ 𝐿𝑋 is not a 𝛼−bounded set. In fact, we suppose that
constant 0.3−net 𝑆 = {𝑥.3 : 𝑥 ∈ 𝑋} in 𝜇. Let 𝑦.2 ∈ 𝑀 (𝐿𝑋), 𝑥 ≠ 𝑦, then 𝑅𝑦.2 = {0𝑋, 𝑥.5}.
Since 𝑆(𝑛) = 𝑥.3 ≤ 𝑥.5 ∈ 𝑅𝑦.2 . So 𝑦.2 is not cluster point of 𝑆. On account of the
arbitrariness of 𝑦 it follows that the 𝑆 has no cluster point in 1𝑋 with height 0.3. Thus 𝜇

is not 𝛼−bounded set.
Now, we show that 𝜇 is 𝑁𝛼−bounded set. Let 𝑆 = {𝑥𝛼 : 𝑥 ∈ 𝑋, 𝛼 ≤ .4} is any

constant 𝛼−net in 𝜇. If 𝛼 < .4 then 𝑅𝑥𝛼 = {0𝑋}, where 𝑐𝑙 (𝑖𝑛𝑡 (0𝑋)) = 0𝑋. Since
𝑆(𝑛) = 𝑥𝛼 ∉ 𝑐𝑙 (𝑖𝑛𝑡 (0𝑋)) = 0𝑋 ∀ 𝑛 ∈ 𝑁, ∀𝛼 < .4. If 𝛼 = .4, then 𝑅𝑥.4 = {0𝑋} and
𝑆(𝑛) = 𝑥𝛼 ∉ 𝑐𝑙 (𝑖𝑛𝑡 (0𝑋)) = 0𝑋. So 𝑥𝛼 is the 𝛿−cluster point of 𝑆 in 1𝑋 1𝑋. Thus 𝜇 is a
𝑁𝛼−bounded set.

Example 3.10 Let 𝑋 = {2, 3, 4, . . .}, 𝐿 = [0, 1],

𝜌𝑛 (𝑥) =
{
0 : 𝑥 = 𝑛

1
2 + 1

𝑛
: 𝑥 ≠ 𝑛 𝑛 ∈ N

𝜂′2 =

{
1 : 𝑥 = 2
1
2 : 𝑥 > 2

𝜎𝑛 (𝑥) = 𝜂′𝑛 (𝑥) =
{
1 : 𝑥 = 𝑛

1
2 − 1

𝑛
: 𝑥 ≠ 𝑛 for 𝑛 ∈ {3, 4, . . .}

Then we have :

(i) 𝑖𝑛𝑡 (𝜌𝑛) = 𝜂𝑛, ∀ 𝑛 ∈ 𝑋.

(ii) 𝑐𝑙 (𝑖𝑛𝑡 (𝜌2)) = 𝜌2 ∨ 𝜎2.

(iii) 𝑐𝑙 (𝑖𝑛𝑡 (𝜌3)) = 𝜌3 ∨ 𝜎2, 𝑐𝑙 (𝑖𝑛𝑡 (𝜌𝑛)) = 𝜌𝑛, ∀ 𝑛 ≥ 4.

We show that 𝜌5 ∈ 𝐿𝑋 is not 𝑁.𝛼–bounded set. Put Ψ = {𝜌𝑛 : 𝑛 ∈ 𝑋}, then then Ψ is
0.8–RF of 1𝑋 where 𝛼 = 0.8 ∈ 𝑀 (𝐿) = (0, 1] (because ∀𝑥 ∈ 𝑋 ∃𝜆 = 𝜌6 ∈ Ψ ∋ 𝑐𝑙 (int(𝜌6)) ∈
𝑅𝑥0.8), where 𝜌6(𝑥) = 0 at 𝑥 = 6 and 𝜌6(𝑥) = 0.6 at 𝑥 ≠ 6.



N. A. Alsaedi / Eur. J. Pure Appl. Math, 18 (4) (2025), 5960 9 of 22

But the family {𝑐𝑙 (int(𝜌𝑛)) : 𝑛 ∈ 𝑋} = {𝜌2 ∨ 𝜎2, 𝜌3 ∨ 𝜎3, 𝜌𝑛 : 𝑛 ≥ 4}. Then any finite
subfamily Ψ◦ = {𝑐𝑙 (int(𝜌𝑛)) : 𝑖 < 𝑛} ∈ 2(Ψ) is not 0.8–RF of 𝜌5 (because ∃𝑥0.8 ∈ 𝜌5 and
∀𝜆 = 𝜌2 we have 𝜌2 ∉ 𝑅𝑥0.8).

Where 𝑐𝑙 (int(𝜌2)) (𝑥) = 𝜌2 ∨ 𝜎2(𝑥) = 1 at 𝑥 = 2 and 𝑐𝑙 (int(𝜌2)) (𝑥) = 𝜌2 ∨ 𝜎2(𝑥) =
1 at 𝑥 ≠ 2. Thus 𝜌5 is not a 𝑁.𝛼–bounded set, however 𝜌5 ∈ 𝑅𝑥0.8 .

Thus 𝑁𝛼𝐵𝑅𝑥0.8 ⊆ 𝑅𝑥0.8 .

Definition 3.11. Let (𝐿𝑋, 𝜏) be an 𝐿 − 𝑡𝑠 and 𝜇 ∈ 𝐿𝑋. Then 𝑥𝛼 ∈ 𝑀 (𝐿𝑋) is
called a 𝑁.𝛼 − 𝑏𝑜𝑢𝑛𝑑𝑒𝑑 adherent point of 𝜇 and write 𝑥𝛼 ∈ 𝑁𝛼𝐵𝑐𝑙 (𝜇) iff 𝜇 ≰ 𝜆 for each
𝜆 ∈ 𝛼𝐵𝑅𝑥𝛼 . If 𝜇 = 𝑁𝛼𝐵𝑐𝑙 (𝜇), then 𝜇 is called a 𝑁𝛼𝐵 − 𝑐𝑙𝑜𝑠𝑒𝑑 𝐿 − 𝑠𝑢𝑏𝑠𝑒𝑡. The family of
all 𝑁𝛼𝐵 − 𝑐𝑙𝑜𝑠𝑒𝑑 𝐿 − 𝑠𝑢𝑏𝑠𝑒𝑡𝑠 is denoted by 𝑁𝛼𝐵𝐶 (𝐿𝑋, 𝜏) and its complement is called
the family of all 𝑁𝛼𝐵 − 𝑜𝑝𝑒𝑛 𝐿 − 𝑠𝑢𝑏𝑠𝑒𝑡𝑠 and denoted by 𝑁𝛼𝐵𝑂 (𝐿𝑋, 𝜏).

Theorem 3.12. Let (𝐿𝑋, 𝜏) be an 𝐿 − 𝑡𝑠 and let 𝜇 ∈ 𝐿𝑋. Then the following
statements are true: (i) 𝜇 ≤ 𝑐𝑙 (𝜇) ≤ 𝑁𝛼𝐵𝑐𝑙 (𝜇). Moreover, 𝑁𝛼𝐵𝑐𝑙 (𝜇) ≤ 𝛼𝐵.𝑐𝑙 (𝜇) [26]

(ii) If 𝜂 ∈ 𝐿𝑋 and 𝜇 ≤ 𝜂 then 𝑁𝛼𝐵𝑐𝑙 (𝜇) ≤ 𝑁𝛼𝐵𝑐𝑙 (𝜂).
(iii) 𝑁𝛼𝐵𝑐𝑙 (𝑁𝛼𝐵𝑐𝑙 (𝜇)) = 𝑁𝛼𝐵𝑐𝑙 (𝜇).
(iv) 𝑁𝛼𝐵𝑐𝑙 (𝜇) = ∧{𝜂 ∈ 𝐿𝑋 : 𝜂 ∈ 𝑁𝛼𝐵𝐶 (𝐿𝑋, 𝜏), 𝜇 ≤ 𝜂}.
Proof. (i) Let 𝑥𝛼 ∈ 𝑀 (𝐿𝑋) such that 𝑥𝛼 ∉ 𝑁𝛼𝐵𝑐𝑙 (𝜇), then there exists 𝜆 ∈ 𝑁𝛼𝐵𝑅𝑥𝛼

such that 𝜇 ≤ 𝜆. Since 𝑁𝛼𝐵𝑅𝑥𝛼 ⊆ 𝑅𝑥𝛼 and so 𝜆 ∈ 𝑅𝑥𝛼 and hence 𝑥𝛼 ∉ 𝑐𝑙 (𝜇). Thus
𝑐𝑙 (𝜇) ≤ 𝑁𝛼𝐵𝑐𝑙 (𝜇).

(ii) Let 𝑥𝛼 ∈ 𝑀 (𝐿𝑋) such that 𝑥𝛼 ∉ 𝑁𝛼𝐵𝑐𝑙 (𝜂), then there exists 𝜆 ∈ 𝑁𝛼𝐵𝑅𝑥𝛼 such
that 𝜂 ≤ 𝜆. Since 𝜇 ≤ 𝜂, then 𝜇 ≤ 𝜆 and so 𝑥𝛼 ∉ 𝑁𝛼𝐵𝑐𝑙 (𝜇). Thus 𝑁𝛼𝐵𝑐𝑙 (𝜇) ≤ 𝑁𝛼𝐵𝑐𝑙 (𝜂).

(iii) Suppose 𝑥𝛼 ∈ 𝑀 (𝐿𝑋) such that 𝑥𝛼 ∈ 𝑁𝛼𝐵𝑐𝑙 (𝑁𝛼𝐵𝑐𝑙 (𝜇)). According to Definition
3.11, we have 𝑁𝛼𝐵𝑐𝑙 (𝜇) ≰ 𝜆 for each 𝜆 ∈ 𝑁𝛼𝐵𝑅𝑥𝛼 . Hence, there exists 𝑦𝛾 ∈ 𝑀 (𝐿𝑋) such
that 𝑦𝛾 ∈ 𝑁𝛼𝐵𝑐𝑙 (𝜇) with 𝑦𝛾 ∉ 𝜆 and so 𝜇 ≰ 𝜆, that is, 𝑥𝛼 ∈ 𝑁𝛼𝐵𝑐𝑙 (𝜇). This shows that
𝑁𝛼𝐵𝑐𝑙 (𝑁𝛼𝐵𝑐𝑙 (𝜇)) ≤ 𝑁𝛼𝐵𝑐𝑙 (𝜇). On the other hand, 𝜇 ≤ 𝑁𝛼𝐵𝑐𝑙 (𝜇) follows from (i) and
so 𝑁𝛼𝐵𝑐𝑙 (𝜇) ≤ 𝑁𝛼𝐵𝑐𝑙 (𝑁𝛼𝐵𝑐𝑙 (𝜇)). Therefore, 𝑁𝛼𝐵𝑐𝑙 (𝑁𝛼𝐵𝑐𝑙 (𝜇)) = 𝑁𝛼𝐵𝑐𝑙 (𝜇).

(iv) On account of (i) and (iii), 𝑁𝛼𝐵𝑐𝑙 (𝜇) is a 𝑁𝛼𝐵 − 𝑐𝑙𝑜𝑠𝑒𝑑 set containing 𝜇, and so
𝑁𝛼𝐵𝑐𝑙 (𝜇) ≥ ∧{𝜂 ∈ 𝐿𝑋 : 𝜂 ∈ 𝑁𝛼𝐵𝐶 (𝐿𝑋, 𝜏), 𝜇 ≤ 𝜂}. Conversely, in case 𝑥𝛼 ∈ 𝑀 (𝐿𝑋) and
𝑥𝛼 ∈ 𝑁𝛼𝐵𝑐𝑙 (𝜇), then 𝜇 ≰ 𝜆 for each 𝜆 ∈ 𝑁𝛼𝐵𝑅𝑥𝛼 . Hence, if 𝜂 is an 𝑁𝛼𝐵 − 𝑐𝑙𝑜𝑠𝑒𝑑 set
containing 𝜇, then 𝜂 ≰ 𝜆, and then 𝑥𝛼 ∈ 𝑁𝛼𝐵𝑐𝑙 (𝜂) = 𝜂. This implies that 𝑁𝛼𝐵𝑐𝑙 (𝜇) ≤
∧{𝜂 ∈ 𝐿𝑋 : 𝜂 ∈ 𝑁𝛼𝐵𝐶 (𝐿𝑋, 𝜏), 𝜇 ≤ 𝜂}. Hence

𝑁𝛼𝐵𝑐𝑙 (𝜇) = ∧{𝜂 ∈ 𝐿𝑋 : 𝜂 ∈ 𝑁𝛼𝐵𝐶 (𝐿𝑋, 𝜏), 𝜇 ≤ 𝜂}
From Theorem 3.12, one can see that every 𝑁𝛼𝐵 − 𝑐𝑙𝑜𝑠𝑒𝑑 𝐿 − 𝑠𝑢𝑏𝑠𝑒𝑡 is a closed

𝐿− 𝑠𝑢𝑏𝑠𝑒𝑡, but the inverse is not true since every closed 𝐿− 𝑠𝑢𝑏𝑠𝑒𝑡 is not a 𝑁.𝛼−𝑏𝑜𝑢𝑛𝑑𝑒𝑑

set in general, as the following example shows.
Example 3.13. By Example 3.10, let 𝜌 ∈ 𝐿𝑋 be a 𝐿–subset, define as follows:

𝜌(𝑥) =
{
1 : 𝑥 = 3, 4, 5, ...
1
6 : 𝑥 = 2

We note that 𝜌 is a closed 𝐿–subset because 𝜌 ∈ 𝜏′ where 𝜏′ is a 𝐿–topology with a
subbase {𝜇′𝑛, 𝜂′𝑛 : 𝑛 ∈ 𝑋}, and we have:



N. A. Alsaedi / Eur. J. Pure Appl. Math, 18 (4) (2025), 5960 10 of 22

𝜂3(𝑥) =
{
0 : 𝑥 ≥ 3
5
6 : 𝑥 < 3

And so

𝜌 = 𝜂′3(𝑥) =
{
1 : 𝑥 ≥ 3
1
6 : 𝑥 < 3

Therefore 𝜌 ∈ 𝜏′. Now, the family Ψ = {𝜇𝑛 : 𝑛 ∈ 𝑋} is 0−cover of 1𝑋. Since
𝛼 = 0 ∈ 𝑃𝑟 (𝐿) = [0, 1) (∀𝑥 ∈ 𝑋 ∃𝜆 ∈ Ψ ∋ 𝜆(𝑥) > 0) which has no finite subfamily Ψ𝑜 of
Ψ such that {𝑖𝑛𝑡 (𝑐𝑙 (𝜇𝑛)) : 𝑖 < 𝑛} is a 0−cover of 𝜌 (since Ψ𝑜 = {𝜇2 ∨ 𝜂2, 𝜇𝑛 : 𝑛 ≥ 4} is not
a 0−cover of 𝜌). Hence 𝜌 is not a 𝑁.𝛼−bounded set.

Theorem 3.14. Let (𝐿𝑋, 𝜏) be an 𝐿−ts. The following statements hold:
(i) 0𝑋, 1𝑋 ∈ 𝑁𝛼𝐵𝐶 (𝐿𝑋, 𝜏).
(ii) If 𝜇1, 𝜇2, ..., 𝜇𝑛 ∈ 𝑁𝛼𝐵𝐶 (𝐿𝑋, 𝜏), then ∨𝑛

𝑖=1 𝜇𝑖 ∈ 𝑁𝛼𝐵𝐶 (𝐿𝑋, 𝜏).
(iii) If {𝜇𝑖 : 𝑖 ∈ 𝐼} ⊆ 𝑁𝛼𝐵𝐶 (𝐿𝑋, 𝜏), then ∧

𝑖∈𝐼 𝜇𝑖 ∈ 𝑁𝛼𝐵𝐶 (𝐿𝑋, 𝜏).
(iv) Every 𝑁𝛼−bounded and closed set is 𝑁𝛼𝐵−closed.
(v) 𝜇 ∈ 𝐿𝑋 is 𝑁𝛼𝐵−closed iff there exists 𝜆 ∈ 𝑁𝛼𝐵𝑅𝑥𝛼 such that 𝜇 ≤ 𝜆 for each 𝑥𝛼 ∈ 𝑀 (𝐿𝑋)
with 𝑥𝛼 ∉ 𝜇.

Proof. (i) Obvious.
(ii) Let 𝜇1, 𝜇2, ..., 𝜇𝑛 ∈ 𝑁𝛼𝐵𝐶 (𝐿𝑋, 𝜏) and 𝑥𝛼 ∈ 𝑀 (𝐿𝑋) such that 𝑥𝛼 ∈ 𝑁𝛼𝐵.𝑐𝑙

(∨𝑛
𝑖=1 𝜇𝑖

)
,

then for each 𝜆 ∈ 𝑁𝛼𝐵𝑅𝑥𝛼 we have
∨𝑛

𝑖=1 𝜇𝑖 ≰ 𝜆 and so 𝜇𝑖 ≰ 𝜆 for some 𝑖 = 1, 2, ..., 𝑛.
Hence 𝑥𝛼 ∈ 𝑁𝛼𝐵.𝑐𝑙 (𝜇𝑖) for some 𝑖 = 1, 2, ..., 𝑛. Since 𝜇𝑖 is a 𝑁𝛼𝐵–closed set, then
𝑁𝛼𝐵.𝑐𝑙 (𝜇𝑖) ≤ 𝜇𝑖 for some 𝑖 = 1, 2, ..., 𝑛 and so 𝑥𝛼 ∈ 𝜇𝑖 for some 𝑖 = 1, 2, ..., 𝑛 and hence
𝑥𝛼 ∈ ∨𝑛

𝑖=1 𝜇𝑖. Thus 𝑁𝛼𝐵.𝑐𝑙
(∨𝑛

𝑖=1 𝜇𝑖
)
≤ ∨𝑛

𝑖=1 𝜇𝑖 . . . (∗)
Conversely, since 𝜇𝑖 ≤ 𝑁𝛼𝐵.𝑐𝑙 (𝜇𝑖) then

∨𝑛
𝑖=1 𝜇𝑖 ≤ 𝑁𝛼𝐵.𝑐𝑙

(∨𝑛
𝑖=1 𝜇𝑖

)
. . . (∗∗).

Hence from (∗) and (∗∗) we have 𝑁𝛼𝐵.𝑐𝑙
(∨𝑛

𝑖=1 𝜇𝑖
)
=
∨𝑛

𝑖=1 𝜇𝑖. Thus
∨𝑛

𝑖=1 𝜇𝑖 ∈ 𝑁𝛼𝐵𝐶 (𝐿𝑋, 𝜏).
(iii) Let 𝜇1, 𝜇2, ..., 𝜇𝑛 ∈ 𝑁𝛼𝐵𝐶 (𝐿𝑋, 𝜏) and 𝑥𝛼 ∈ 𝑀 (𝐿𝑋) such that 𝑥𝛼 ∈ 𝑁𝛼𝐵.𝑐𝑙 (∧𝑖∈𝐼 𝜇𝑖),

then for each 𝜆 ∈ 𝑁𝛼𝐵𝑅𝑥𝛼 we have
∧

𝑖∈𝐼 𝜇𝑖 ≰ 𝜆 and so 𝜇𝑖 ≰ 𝜆 for each 𝑖 ∈ 𝐼. Hence
𝑥𝛼 ∈ 𝑁𝛼𝐵.𝑐𝑙 (𝜇𝑖) for each 𝑖 ∈ 𝐼. Since 𝜇𝑖 is a 𝑁𝛼𝐵–closed set, then 𝑁𝛼𝐵.𝑐𝑙 (𝜇𝑖) ≤ 𝜇𝑖 for
each 𝑖 ∈ 𝐼 and so 𝑥𝛼 ∈ 𝜇𝑖 for each 𝑖 ∈ 𝐼 and hence 𝑥𝛼 ∈ ∧

𝑖∈𝐼 𝜇𝑖. Thus 𝑁𝛼𝐵.𝑐𝑙 (∧𝑖∈𝐼 𝜇𝑖) ≤∧
𝑖∈𝐼 𝜇𝑖 . . . (∗).

Conversely, since 𝜇𝑖 ≤ 𝑁𝛼𝐵.𝑐𝑙 (𝜇𝑖) then
∧

𝑖∈𝐼 𝜇𝑖 ≤ 𝑁𝛼𝐵.𝑐𝑙 (∧𝑖∈𝐼 𝜇𝑖) . . . (∗∗).
Hence from (∗) and (∗∗) we have 𝑁𝛼𝐵.𝑐𝑙 (∧𝑖∈𝐼 𝜇𝑖) =

∧
𝑖∈𝐼 𝜇𝑖. Thus

∧
𝑖∈𝐼 𝜇𝑖 ∈ 𝑁𝛼𝐵𝐶 (𝐿𝑋, 𝜏).

(iv) Let 𝜇 ∈ 𝐿𝑋 be a 𝑁𝛼−bounded and closed set and let 𝑥𝛼 ∈ 𝑀 (𝐿𝑋) such that
𝑥𝛼 ∉ 𝜇, since 𝜇 is a 𝑁𝛼−bounded and closed set, then 𝜇 ∈ 𝑁𝛼𝐵𝑅𝑥𝛼 . Since 𝜇 ≤ 𝜇, then
𝑥𝛼 ∉ 𝑁𝛼𝐵𝑐𝑙 (𝜇) and so 𝑁𝛼𝐵𝑐𝑙 (𝜇) ≤ 𝜇. Therefore 𝜇 is 𝑁𝛼𝐵−closed set.

(v) Suppose that 𝜇 is a 𝑁𝛼𝐵−closed set, 𝑥𝛼 ∈ 𝑀 (𝐿𝑋) and 𝑥𝛼 ∉ 𝜇. By Definition 3.11,
there exists 𝜆 ∈ 𝑁𝛼𝐵𝑅𝑥𝛼 with 𝜇 ≤ 𝜆. Conversely, provided that the condition is satisfied.
If 𝜇 is not a 𝑁𝛼𝐵−closed set, then there exists 𝑥𝛼 ∈ 𝑀 (𝐿𝑋) such that 𝑥𝛼 ∈ 𝑁𝛼𝐵𝑐𝑙 (𝜇)
and 𝑥𝛼 ∉ 𝜇. Hence 𝜇 ≰ 𝜆 for each 𝜆 ∈ 𝑁𝛼𝐵𝑅𝑥𝛼 . It conflicts with the hypothesis, and so
𝜇 is a 𝑁𝛼𝐵−closed set.



N. A. Alsaedi / Eur. J. Pure Appl. Math, 18 (4) (2025), 5960 11 of 22

Theorem 3.15. Let (𝐿𝑋, 𝜏) be an 𝐿 − 𝑡𝑠 and 𝜇 ∈ 𝐿𝑋. Then the mapping 𝑁𝛼𝐵𝑐𝑙 :
𝐿𝑋 → 𝐿𝑋 is called a closure operator of 𝑁𝛼−boundedness iff it satisfies:
(i) 𝑁𝛼𝐵𝑐𝑙 (0𝑋) = 0𝑋.
(ii) 𝜇 ≤ 𝑁𝛼𝐵𝑐𝑙 (𝜇).
(iii) 𝑁𝛼𝐵𝑐𝑙 (𝜇 ∨ 𝜂) = 𝑁𝛼𝐵𝑐𝑙 (𝜇) ∨ 𝑁𝛼𝐵𝑐𝑙 (𝜂).
(iv) 𝑁𝛼𝐵𝑐𝑙 (𝑁𝛼𝐵𝑐𝑙 (𝜇)) = 𝑁𝛼𝐵𝑐𝑙 (𝜇).

A closure operator of 𝑁𝛼−boundedness 𝑁𝛼𝐵𝑐𝑙 generates 𝐿−topology 𝜏𝑁𝛼𝐵𝑐𝑙 on 𝐿𝑋

as: 𝜏𝑁𝛼𝐵𝑐𝑙 = {𝜇 ∈ 𝐿𝑋 : 𝑁𝛼𝐵𝑐𝑙 (𝜇′) = 𝜇′}.
Proof. It follows directly from Theorems 3.12 and 3.14.
Theorem 3.16. Let (𝐿𝑋, 𝜏) be an 𝐿 − 𝑡𝑠. Then:

(i) 𝜏𝛼𝐵 ≤ 𝜏𝑁𝛼𝐵 ≤ 𝜏.
(ii) If (𝐿𝑋, 𝜏) is 𝛼−bounded (resp. 𝑁𝛼−bounded space), then 𝜏 = 𝜏𝛼𝐵 (resp. 𝜏 = 𝜏𝑁𝛼𝐵).
(iii) If (𝐿𝑋, 𝜏) is 𝐿𝑅2−space, then 𝜏𝛼𝐵 = 𝜏𝑁𝛼𝐵.

Proof. (i) Let 𝜇 ∈ 𝜏𝛼𝐵, then 𝛼𝐵𝑐𝑙 (𝜇′) ≤ 𝜇′. Since 𝑁𝛼𝐵𝑐𝑙 (𝜇′) ≤ 𝛼𝐵𝑐𝑙 (𝜇′) hence
𝑁𝛼𝐵𝑐𝑙 (𝜇′) ≤ 𝜇′ and so 𝜇 ∈ 𝜏𝑁𝛼𝐵. If 𝜇 ∈ 𝜏𝑁𝛼𝐵, then 𝑁𝛼𝐵𝑐𝑙 (𝜇′) ≤ 𝜇′ and so 𝜇 ∈ 𝜏. also
if 𝜇 ∈ 𝜏𝑁𝛼𝐵, then 𝑁𝛼𝐵𝑐𝑙 (𝜇′) ≤ 𝜇′. Since 𝑐𝑙 (𝜇′) ≤ 𝑁𝐵𝑐𝑙 (𝜇′), hence 𝑐𝑙 (𝜇′) ≤ 𝜇′ and so
𝜇 ∈ 𝜏.
(ii) We note that 𝜏𝛼𝐵 ≤ 𝜏 from (i). Now, let 𝜇 ∈ 𝜏 then 𝜇′ ∈ 𝜏′. Since 1𝑋 is a 𝑁𝛼−bounded
and 𝜇′ ≤ 1𝑋, 𝜇

′ is 𝑁𝛼−bounded. By Theorem 3.7 and by Theorem 3.14 (iv), we have 𝜇′

that is a 𝑁𝛼𝐵−closed set and so 𝜇′ ∈ 𝜏𝑁𝛼𝐵. Thus 𝜏 = 𝜏𝑁𝛼𝐵.
Definition 3.17. Let (𝐿𝑋, 𝜏) be an 𝐿 − 𝑡𝑠, 𝜇 ∈ 𝐿𝑋 and 𝑁𝛼𝐵.𝑖𝑛𝑡 (𝜇) = ∨{𝜌 ∈ 𝐿𝑋 : 𝜌 ∈

𝑁𝛼𝐵𝑂 (𝐿𝑋, 𝜏), 𝜌 ≤ 𝜇}. We say that 𝑁𝛼𝐵.𝑖𝑛𝑡 (𝜇) is the 𝑁𝛼𝐵−interior of 𝜇.
The following Theorem shows the relationships between 𝑁𝛼𝐵−closure operator and

𝑁𝛼𝐵−interior operator.
Theorem 3.18. Let (𝐿𝑋, 𝜏) be an 𝐿 − 𝑡𝑠 and 𝜇 ∈ 𝐿𝑋. Then the following are true:

(i) 𝜇 is 𝑁𝛼𝐵−open iff 𝜇 = 𝑁𝛼𝐵.𝑖𝑛𝑡 (𝜇).
(ii) (𝑁𝛼𝐵𝑐𝑙 (𝜇))′ = 𝑁𝛼𝐵.𝑖𝑛𝑡 (𝜇′) and (𝑁𝛼𝐵.𝑖𝑛𝑡 (𝜇))′ = 𝑁𝛼𝐵𝑐𝑙 (𝜇′).
(iii) 𝑁𝛼𝐵𝑐𝑙 (𝜇) = (𝑁𝛼𝐵.𝑖𝑛𝑡 (𝜇′))′ and 𝑁𝛼𝑖𝑛𝑡 (𝜇) = (𝑁𝛼𝑐𝑙 (𝜇′))′.
(iv) 𝑁𝛼𝐵.𝑖𝑛𝑡 (𝜇) ≤ 𝛼𝐵.𝑖𝑛𝑡 (𝜇) ≤ 𝑖𝑛𝑡 (𝜇) ≤ 𝜇.
(v) If 𝜂 ∈ 𝐿𝑋 and 𝜇 ≤ 𝜂 then 𝑁𝛼𝐵.𝑖𝑛𝑡 (𝜇) ≤ 𝑁𝛼𝐵.𝑖𝑛𝑡 (𝜂).
(vi) 𝑁𝛼𝐵.𝑖𝑛𝑡 (𝑁𝛼𝐵.𝑖𝑛𝑡 (𝜇)) = 𝑁𝛼𝐵.𝑖𝑛𝑡 (𝜇).

Proof. (i) Let 𝜇 ∈ 𝐿𝑋 be an 𝑁𝛼𝐵− open set, then 𝑁𝛼𝐵.𝑖𝑛𝑡 (𝜇) = ∨{𝜌 ∈ 𝐿𝑋 : 𝜌 ∈
𝑁𝛼𝐵𝑂 (𝐿𝑋, 𝜏), 𝜌 ≤ 𝜇} = 𝜇 and so 𝜇 = 𝑁𝛼𝐵.𝑖𝑛𝑡 (𝜇).

Conversely, let 𝜇 = 𝑁𝛼𝐵.𝑖𝑛𝑡 (𝜇), since 𝑁𝛼𝐵.𝑖𝑛𝑡 (𝜇) = ∨{𝜌 ∈ 𝐿𝑋 : 𝜌 ∈ 𝑁𝛼𝐵𝑂 (𝐿𝑋, 𝜏), 𝜌 ≤
𝜇} = 𝜇. Therefore 𝜇 is 𝑁𝛼𝐵−open set.

(ii) It follows directly from Theorem 3.12 (iv) and Definition 3.17.
(iii) It follows directly from (ii).
(iv) It follows directly from (ii) and Theorems 3.12 (i).
(v) It follows directly from (ii) and Theorem 3.12 (ii).
(vi) It follows directly from (ii) and Theorem 3.12 (iii).

Theorem 3.19. Let (𝐿𝑋, 𝜏) be an 𝐿 − 𝑡𝑠. The following statements hold:
(i) 0𝑋, 1𝑋 ∈ 𝑁𝛼𝐵𝑂 (𝐿𝑋, 𝜏).
(ii) If 𝜇1, 𝜇2, ..., 𝜇𝑛 ∈ 𝑁𝛼𝐵𝑂 (𝐿𝑋, 𝜏), then ∧𝑛

𝑖=1 𝜇𝑖 ∈ 𝑁𝛼𝐵𝑂 (𝐿𝑋, 𝜏).



N. A. Alsaedi / Eur. J. Pure Appl. Math, 18 (4) (2025), 5960 12 of 22

(iii) If {𝜇𝑖 : 𝑖 ∈ 𝐼} ⊆ 𝑁𝛼𝐵𝑂 (𝐿𝑋, 𝜏), then ∨
𝑖∈𝐼 𝜇𝑖 ∈ 𝑁𝛼𝐵𝑂 (𝐿𝑋, 𝜏).

Proof. It is similar to the proof of Theorem 3.14.

Definition 3.20. Let (𝐿𝑋, 𝜏) be an 𝐿–ts and 𝑆 be a molecular net in 𝐿𝑋. Then
𝑥𝛼 ∈ 𝑀 (𝐿𝑋) is called a 𝑁𝛼–bounded limit point of 𝑆, (or 𝑆 𝑁𝛼𝐵–converges to 𝑥𝛼) in

symbol 𝑆
𝑁𝛼𝐵−−−−→ 𝑥𝛼 if for every 𝜇 ∈ 𝑁𝛼𝐵𝑅𝑥𝛼 and there is 𝑛 ∈ 𝐷 such that 𝑚 ∈ 𝐷 and

𝑚 ≥ 𝑛 we have 𝑆(𝑚) ∉ 𝜇.
The union of all 𝑁𝛼–bounded limit points of 𝑆 is denoted by 𝑁𝛼𝐵. lim(𝑆).

Definition 3.21. Let (𝐿𝑋, 𝜏) be an 𝐿–ts and 𝑆 be a molecular net in 𝐿𝑋. Then

𝑥𝛼 ∈ 𝑀 (𝐿𝑋) is called a 𝑁𝛼–bounded cluster point of 𝑆, in symbol 𝑆
𝑁𝛼𝐵∝ 𝑥𝛼, if for every

𝜇 ∈ 𝑁𝛼𝐵𝑅𝑥𝛼 and every 𝑛 ∈ 𝐷 there is 𝑚 ∈ 𝐷 such that 𝑚 ≥ 𝑛 and 𝑆(𝑚) ∉ 𝜇.
The union of all 𝑁𝛼–bounded cluster points of 𝑆 is denoted by 𝑁𝛼𝐵.𝑎𝑑ℎ(𝑆).

Theorem 3.22. Suppose that 𝑆 is a molecular net in (𝐿𝑋, 𝜏), 𝜇 ∈ 𝐿𝑋 and 𝑥𝛼 ∈ 𝑀 (𝐿𝑋).
Then the following statements hold:

(i) 𝑥𝛼 ∈ 𝑁𝛼𝐵. lim(𝑆) iff 𝑆
𝑁𝛼𝐵−−−−→ 𝑥𝛼.

(ii) 𝑥𝛼 ∈ 𝑁𝛼𝐵.𝑎𝑑ℎ(𝑆) iff 𝑆
𝑁𝛼𝐵∝ 𝑥𝛼.

(iii) lim(𝑆) ≤ 𝑁𝛼𝐵. lim(𝑆).

(iv) 𝑎𝑑ℎ(𝑆) ≤ 𝑁𝛼𝐵.𝑎𝑑ℎ(𝑆).

(v) 𝑁𝛼𝐵. lim(𝑆) and 𝑁𝛼𝐵.𝑎𝑑ℎ(𝑆) are 𝑁𝛼𝐵–closed in 𝐿𝑋.

(vi) 𝑥𝛼 ∈ 𝑁𝛼𝐵.𝑐𝑙 (𝜇) (resp. 𝑥𝛼 ∈ 𝛿𝑐𝑙 (𝜇) [9]), iff there exists a molecular net 𝑆 in 𝜇 such
that 𝑆 is 𝑁𝛼𝐵–converges (resp. 𝛿–converges) to 𝑥𝛼.

Proof. (i) Let 𝑥𝛼 ∈ 𝑁𝛼𝐵. lim(𝑆) and let 𝜆 ∈ 𝑁𝛼𝐵𝑅𝑥𝛼 . Since 𝑥𝛼 ∉ 𝜆, then
𝑁𝛼𝐵. lim(𝑆) ∉ 𝜆. Therefore there exists 𝑦𝛾 ∈ 𝑀 (𝐿𝑋) such that 𝑦𝛾 ∈ 𝑁𝛼𝐵. lim(𝑆)
and 𝑦𝛾 ∉ 𝜆. Then 𝜆 ∈ 𝑁𝛼𝐵𝑅𝑦𝛾 , and so there is 𝑛 ∈ 𝐷 such that for each 𝑚 ∈ 𝐷 and 𝑚 ≥ 𝑛

we have 𝑆(𝑚) ∉ 𝜆, but since 𝜆 ∈ 𝑁𝛼𝐵𝑅𝑥𝛼 so 𝑆
𝑁𝛼𝐵−−−−→ 𝑥𝛼. Conversely, let 𝑆

𝑁𝛼𝐵−−−−→ 𝑥𝛼, then
by Definition 3.20, we have 𝑥𝛼 ∈ 𝑁𝛼𝐵. lim(𝑆).

(ii) Let 𝑥𝛼 ∈ 𝑁𝛼𝐵.(𝑆) and let 𝜆 ∈ 𝑁𝛼𝐵𝑅𝑥𝛼 . Since 𝑥𝛼 ∈ 𝑁𝛼𝐵.(𝑆), then every 𝑛 ∈ 𝐷

there is 𝑚 ∈ 𝐷 such that 𝑚 ≥ 𝑛 and 𝑆(𝑚) ∉ 𝜆, hence 𝑆
𝑁𝛼𝐵∝ 𝑥𝛼. Conversely, let 𝑆

𝑁𝛼𝐵∝ 𝑥𝛼,
then by Definition 3.21 we have 𝑥𝛼 ∈ 𝑁𝛼𝐵.(𝑆).
(iii) Let 𝑥𝛼 ∈ lim(𝑆) and let 𝜂 ∈ 𝑁𝛼𝐵𝑅𝑥𝛼 . Since 𝑁𝛼𝐵𝑅𝑥𝛼 ⊆ 𝑅𝑥𝛼 , then 𝜂 ∈ 𝑅𝑥𝛼 . And since
𝑥𝛼 ∈ lim(𝑆), then, for each 𝜆 ∈ 𝑅𝑥𝛼 there is 𝑛 ∈ 𝐷 such that for each 𝑚 ∈ 𝐷 and 𝑚 ≥ 𝑛,
we have 𝑆(𝑚) ∉ 𝜆 and so 𝑆(𝑚) ∉ 𝜂. Hence, 𝑥𝛼 ∈ 𝑁𝛼𝐵. lim(𝑆). So lim(𝑆) ≤ 𝑁𝛼𝐵. lim(𝑆).

(iv) Let 𝑥𝛼 ∈ 𝑎𝑑ℎ(𝑆) and let 𝜂 ∈ 𝑁𝛼𝐵𝑅𝑥𝛼 . Since 𝑁𝛼𝐵𝑅𝑥𝛼 ⊆ 𝑅𝑥𝛼 , then 𝜂 ∈ 𝑅𝑥𝛼 . And
since 𝑥𝛼 ∈ 𝑎𝑑ℎ(𝑆), then, for each 𝜆 ∈ 𝑅𝑥𝛼 and for each 𝑛 ∈ 𝐷 there exists 𝑚 ∈ 𝐷 such that
𝑚 ≥ 𝑛 we have 𝑆(𝑚) ∉ 𝜆. And so 𝑆(𝑚) ∉ 𝜂. Hence 𝑥𝛼 ∈ 𝑁𝛼𝐵.(𝑆). So 𝑎𝑑ℎ(𝑆) ≤ 𝑁𝛼𝐵.(𝑆).
(v) Let 𝑥𝛼 ∈ 𝑁𝛼𝐵.𝑐𝑙 (𝑁𝛼𝐵. lim(𝑆)), then 𝑁𝛼𝐵. lim(𝑆) ≰ 𝜆 for each 𝜆 ∈ 𝑁𝛼𝐵𝑅𝑥𝛼 and
then there exists 𝑦𝛾 ∈ 𝑀 (𝐿𝑋) such that 𝑦𝛾 ∈ 𝑁𝛼𝐵. lim(𝑆) and 𝑦𝛾 ∉ 𝜆. Then for each



N. A. Alsaedi / Eur. J. Pure Appl. Math, 18 (4) (2025), 5960 13 of 22

𝜇 ∈ 𝑁𝛼𝐵𝑅𝑦𝛾 , there is 𝑛 ∈ 𝐷 such that for each 𝑚 ∈ 𝐷 and 𝑚 ≥ 𝑛 we have 𝑆(𝑚) ∉ 𝜇, and
so 𝑆(𝑚) ∉ 𝜆. Hence 𝑥𝛼 ∈ 𝑁𝛼𝐵. lim(𝑆). Thus 𝑁𝛼𝐵.𝑐𝑙 (𝑁𝛼𝐵. lim(𝑆)) ≤ 𝑁𝛼𝐵. lim(𝑆) and so
𝑁𝛼𝐵. lim(𝑆) is a 𝑁𝛼𝐵–closed set.

Similarly, one can easily verify that 𝑁𝛼𝐵.𝑐𝑙 (𝑁𝛼𝐵.(𝑆)) ≤ 𝑁𝛼𝐵.(𝑆).
(vi) Let 𝑥𝛼 ∈ 𝑀 (𝐿𝑋) such that 𝑥𝛼 ∈ 𝑁𝛼𝐵.𝑐𝑙 (𝜇), then 𝜇 ≰ 𝜆 for each 𝜆 ∈ 𝑁𝛼𝐵𝑅𝑥𝛼 . Since
𝜇 ≰ 𝜆 then, there exists 𝛼(𝜇, 𝜆) ∈ 𝑀 (𝐿) such that 𝑥𝛼(𝜇,𝜆) ∈ 𝜇 with 𝑥𝛼(𝜇,𝜆) ∉ 𝜆. Since the
pair (𝑁𝛼𝐵𝑅𝑥𝛼 , ≥) is a directed set so we can define a molecular net 𝑆 : 𝑁𝛼𝐵𝑅𝑥𝛼 → 𝑀 (𝐿𝑋)
as follows 𝑆(𝜆) = 𝑥𝛼(𝜇,𝜆) for each 𝜆 ∈ 𝑁𝛼𝐵𝑅𝑥𝛼 . Hence 𝑆 is a molecular net in 𝜇.

Now let 𝜂 ∈ 𝑁𝛼𝐵𝑅𝑥𝛼 such that 𝜆 ≤ 𝜂, so we have there exists 𝑆(𝜂) = 𝑥𝛼(𝜇,𝜂) ∉ 𝜂 and
so 𝑆(𝜂) = 𝑥𝛼(𝜇,𝜂) ∉ 𝜆. Hence 𝑆 is 𝑁𝛼𝐵–converges to 𝑥𝛼.

Conversely, let 𝑆 be a molecular net in 𝜇 such that 𝑆 is 𝑁𝛼𝐵–converges to 𝑥𝛼, then
for each 𝜆 ∈ 𝑁𝛼𝐵𝑅𝑥𝛼 there is 𝑛 ∈ 𝐷 such for each 𝑚 ∈ 𝐷 and 𝑚 ≥ 𝑛, we have 𝑆(𝑚) ∉ 𝜆.
Since 𝑆(𝑛) ∈ 𝜇 for each 𝑛 ∈ 𝐷, 𝑚 ∈ 𝐷. So 𝑆(𝑚) ∈ 𝜇 and 𝜇 ≥ 𝑆(𝑚) > 𝜆 hence 𝜇 ≰ 𝜆 for
each 𝜆 ∈ 𝑁𝛼𝐵𝑅𝑥𝛼 . This means that 𝑥𝛼 ∈ 𝑁𝛼𝐵.𝑐𝑙 (𝜇).

Definition 3.23. Let (𝐿𝑋, 𝜏) be an 𝐿–ts and 𝐼 be an ideal in 𝐿𝑋. Then 𝑥𝛼 ∈ 𝑀 (𝐿𝑋) is
called:
(i) limit point of 𝐼 [25], (or 𝐼 converges to 𝑥𝛼) in symbol 𝐼 → 𝑥𝛼 if 𝑅𝑥𝛼 ⊆ 𝐼. The union of
all limit points of 𝐼 is denoted by lim(𝐼).
(ii) 𝑁𝛼𝐵–bounded limit point of 𝐼, (or 𝐼 𝑁𝛼𝐵–converges to 𝑥𝛼) in symbol 𝐼

𝑁𝛼𝐵−−−−→ 𝑥𝛼 if
𝑁𝛼𝐵𝑅𝑥𝛼 ⊆ 𝐼. The union of all 𝑁𝛼𝐵–bounded limit points of 𝐼 is denoted by 𝑁𝛼𝐵. lim(𝐼).

Definition 3.24. Let (𝐿𝑋, 𝜏) be an 𝐿–ts and 𝐼 be an ideal in 𝐿𝑋. Then 𝑥𝛼 ∈ 𝑀 (𝐿𝑋) is
called:
(i) Cluster point of 𝐼 [25], in symbol 𝐼 ∝ 𝑥𝛼 if for every 𝜇 ∈ 𝑅𝑥𝛼 and every 𝜆 ∈ 𝐼, 𝜆∨𝜇 ≠ 1𝑋.
The union of all cluster points of 𝐼 is denoted by adh(𝐼).
(ii) 𝑁𝛼𝐵–bounded cluster point of 𝐼, in symbol 𝐼

𝑁𝛼𝐵∝ 𝑥𝛼 if for every 𝜇 ∈ 𝑁𝛼𝐵𝑅𝑥𝛼 and
every 𝜆 ∈ 𝐼, 𝜆 ∨ 𝜇 ≠ 1𝑋. The union of all 𝑁𝛼𝐵–bounded cluster points of 𝐼 is denoted by
𝑁𝛼𝐵. adh(𝐼).

Theorem 3.25. Suppose that 𝐼 is an ideal in (𝐿𝑋, 𝜏), 𝜇 ∈ 𝐿𝑋 and 𝑥𝛼 ∈ 𝑀 (𝐿𝑋).
Then the following statements hold:

(i) 𝑥𝛼 ∈ 𝑁𝛼𝐵. lim(𝐼) iff 𝐼
𝑁𝛼𝐵−−−−→ 𝑥𝛼.

(ii) 𝑥𝛼 ∈ 𝑁𝛼𝐵.adh(𝐼) iff 𝐼
𝑁𝛼𝐵∝ 𝑥𝛼.

(iii) lim(𝐼) ≤ 𝑁𝛼𝐵. lim(𝐼) ≤ 𝛼𝐵. lim(𝐼).

(iv) adh(𝐼) ≤ 𝑁𝛼𝐵.adh(𝐼) ≤ 𝛼𝐵.adh(𝐼).

(v) 𝑥𝛼 ∈ 𝑁𝛼𝐵.𝑐𝑙 (𝜇) iff there exists an ideal 𝐼 in 𝐿𝑋 such that 𝐼
𝑁𝛼𝐵−−−−→ 𝑥𝛼 and 𝜇 ∉ 𝐼.

(vi) 𝑁𝛼𝐵. lim(𝐼) and 𝑁𝛼𝐵.adh(𝐼) are 𝑁𝛼𝐵–closed set in 𝐿𝑋.



N. A. Alsaedi / Eur. J. Pure Appl. Math, 18 (4) (2025), 5960 14 of 22

Proof. (i) Let 𝑥𝛼 ∈ 𝑁𝛼𝐵. lim(𝐼) and let 𝜆 ∈ 𝑁𝛼𝐵𝑅𝑥𝛼 . Since 𝑥𝛼 ∉ 𝜆 and 𝑥𝛼 ∈ 𝑁𝛼𝐵. lim(𝐼),
then 𝑁𝛼𝐵. lim(𝐼) ≰ 𝜆. Therefore there exists 𝑦𝛾 ∈ 𝑀 (𝐿𝑋) such that 𝑦𝛾 ∈ 𝑁𝛼𝐵. lim(𝐼)
and 𝑦𝛾 ∉ 𝜆. Then 𝜆 ∈ 𝑁𝛼𝐵𝑅𝑦𝛾 and so 𝑁𝛼𝐵𝑅𝑥𝛼 ⊆ 𝑁𝛼𝐵𝑅𝑦𝛾 ⊆ 𝐼 hence 𝑁𝛼𝐵𝑅𝑥𝛼 ⊆ 𝐼.

Thus 𝐼
𝑁𝛼𝐵−−−−→ 𝑥𝛼. Conversely, let 𝐼

𝑁𝛼𝐵−−−−→ 𝑥𝛼, then by Definition 3.23 (ii) we have
𝑥𝛼 ∈ 𝑁𝛼𝐵. lim(𝐼).

(ii) Let 𝑥𝛼 ∈ 𝑁𝛼𝐵.𝑎𝑑ℎ(𝐼) and let 𝜆 ∈ 𝑁𝛼𝐵𝑅𝑥𝛼 . Since 𝑥𝛼 ∉ 𝜆 and 𝑥𝛼 ∈ 𝑁𝛼𝐵.𝑎𝑑ℎ(𝐼),
therefore 𝑁𝛼𝐵.𝑎𝑑ℎ(𝐼) ≰ 𝜆 and so there exists 𝑦𝛾 ∈ 𝑀 (𝐿𝑋) such that 𝑦𝛾 ∈ 𝑁𝛼𝐵.𝑎𝑑ℎ(𝐼)
and 𝑦𝛾 ∉ 𝜆 hence 𝜆 ∈ 𝑁𝛼𝐵𝑅𝑦𝛾 and so 𝑁𝛼𝐵𝑅𝑥𝛼 ⊆ 𝑁𝛼𝐵𝑅𝑦𝛾 and 𝜇 ∨ 𝜆 ≠ 1𝑋 for each

𝜇 ∈ 𝐼 hence 𝐼
𝑁𝛼𝐵∝ 𝑥𝛼. Conversely, let 𝐼

𝑁𝛼𝐵∝ 𝑥𝛼, then by Definition 3.24 (ii) we have
𝑥𝛼 ∈ 𝑁𝛼𝐵.𝑎𝑑ℎ(𝐼).

(iii) Let 𝑥𝛼 ∈ lim(𝐼) and let 𝜂 ∈ 𝑁𝛼𝐵𝑅𝑥𝛼 . Since 𝑁𝛼𝐵𝑅𝑥𝛼 ⊆ 𝑅𝑥𝛼 , then 𝜂 ∈ 𝑅𝑥𝛼 . And
since 𝑥𝛼 ∈ lim(𝐼), then 𝑅𝑥𝛼 ⊆ 𝐼 so for each 𝜂 ∈ 𝑅𝑥𝛼 , 𝜂 ∈ 𝐼 and since 𝜂 ∈ 𝑁𝛼𝐵𝑅𝑥𝛼 , so
𝑁𝛼𝐵𝑅𝑥𝛼 ⊆ 𝐼. Hence 𝑥𝛼 ∈ 𝑁𝛼𝐵. lim(𝐼). So lim(𝐼) ≤ 𝑁𝛼𝐵. lim(𝐼).

Let 𝑥𝛼 ∈ 𝑁𝛼𝐵. lim(𝐼) and let 𝜂 ∈ 𝛼𝐵𝑅𝑥𝛼 . Since 𝛼𝐵𝑅𝑥𝛼 ⊆ 𝑁𝛼𝐵𝑅𝑥𝛼 , then 𝜂 ∈ 𝑁𝛼𝐵𝑅𝑥𝛼 .
And since 𝑥𝛼 ∈ 𝑁𝛼𝐵. lim(𝐼), then 𝑁𝛼𝐵𝑅𝑥𝛼 ⊆ 𝐼 so for each 𝜂 ∈ 𝑁𝛼𝐵𝑅𝑥𝛼 , 𝜂 ∈ 𝐼 and
since 𝜂 ∈ 𝛼𝐵𝑅𝑥𝛼 , 𝜂 ∈ 𝐼 and since 𝜂 ∈ 𝛼𝐵𝑅𝑥𝛼 , so 𝛼𝐵𝑅𝑥𝛼 ⊆ 𝐼. Hence 𝑥𝛼 ∈ 𝛼𝐵. lim(𝐼). So
𝑁𝛼𝐵. lim(𝐼) ≤ 𝛼𝐵. lim(𝐼).

(iv) Let 𝑥𝛼 ∈ adh(𝐼) and let 𝜂 ∈ 𝑁𝛼𝐵𝑅𝑥𝛼 . Since 𝑥𝛼 ∈ adh(𝐼), so for each 𝜆 ∈ 𝑅𝑥𝛼 , 𝜆 ∈ 𝐼

and since 𝜂 ∈ 𝑁𝛼𝐵𝑅𝑥𝛼 so 𝜂 ∈ 𝑅𝑥𝛼 . Hence 𝑥𝛼 ∈ 𝑁𝛼𝐵.adh(𝐼). So adh(𝐼) ≤ 𝛼𝐵.adh(𝐼).
Similarly, one can easily verify that 𝑁𝛼𝐵.adh(𝐼) ≤ 𝛼𝐵.adh(𝐼).

(v) Let 𝑥𝛼 ∈ 𝑀 (𝐿𝑋) such that 𝑥𝛼 ∈ 𝑁𝛼𝐵.𝑐𝑙 (𝜇). The family 𝐼 = {𝜌 ∈ 𝐿𝑋 : ∃𝜆 ∈
𝑁𝛼𝐵𝑅𝑥𝛼 ∋ 𝜌 ≤ 𝜆} is an ideal in 𝐿𝑋. Now we show that 𝜇 ∉ 𝐼. Since 𝑥𝛼 ∈ 𝑁𝛼𝐵.𝑐𝑙 (𝜇),
then for each 𝜆 ∈ 𝑁𝛼𝐵𝑅𝑥𝛼 , 𝜇 ≰ 𝜆. So by definition of 𝐼 we have 𝜇 ∉ 𝐼. Finally, we

show that 𝐼
𝑁𝛼𝐵−−−−→ 𝑥𝛼. Let 𝜆 ∈ 𝑁𝛼𝐵𝑅𝑥𝛼 , since 𝜆 ≤ 𝜆, then 𝜆 ∈ 𝐼. So 𝑁𝛼𝐵𝑅𝑥𝛼 ⊆ 𝐼. Thus

𝐼
𝑁𝛼𝐵−−−−→ 𝑥𝛼.

Conversely, let 𝐼 be an ideal in 𝐿𝑋 such that 𝐼
𝑁𝛼𝐵−−−−→ 𝑥𝛼 and 𝜇 ∉ 𝐼. Then for each

𝜆 ∈ 𝑁𝛼𝐵𝑅𝑥𝛼 , 𝜆 ∈ 𝐼. Since 𝜆 ∈ 𝐼, 𝜇 ∉ 𝐼, 𝜇 ≰ 𝜆 and so 𝑥𝛼 ∈ 𝑁𝛼𝐵.𝑐𝑙 (𝜇).
(vi) Let 𝑥𝛼 ∈ 𝑁𝛼𝐵.𝑐𝑙 (𝑁𝛼𝐵. lim(𝐼)), then 𝑁𝛼𝐵. lim(𝐼) ≰ 𝜆 for each 𝜆 ∈ 𝑁𝛼𝐵𝑅𝑥𝛼

and then there exists 𝑦𝛾 ∈ 𝑀 (𝐿𝑋) such that 𝑦𝛾 ∈ 𝑁𝛼𝐵. lim(𝐼) and 𝑦𝛾 ∉ 𝜆. Since

𝜆 ∈ 𝑁𝛼𝐵𝑅𝑦𝛾 and 𝐼
𝑁𝛼𝐵−−−−→ 𝑥𝛼 then 𝜂 ∈ 𝐼 for each 𝜂 ∈ 𝑁𝛼𝐵𝑅𝑥𝛼 . Since 𝑦𝛾 ∉ 𝜆 then

𝜆 ∈ 𝐼. But 𝜆 ∈ 𝑁𝛼𝐵𝑅𝑥𝛼 and so 𝑥𝛼 ∈ 𝑁𝛼𝐵. lim(𝐼). Thus 𝑁𝛼𝐵.𝑐𝑙 (𝑁𝛼𝐵. lim(𝐼)) ≤
𝑁𝛼𝐵. lim(𝐼) and so 𝑁𝛼𝐵. lim(𝐼) is a 𝑁𝛼𝐵–closed set. Similarly, one can easily verify that
𝑁𝛼𝐵.𝑐𝑙 (𝑁𝛼𝐵.adh(𝐼)) ≤ 𝑁𝛼𝐵.adh(𝐼).

Theorem 3.26. Suppose that 𝑆 is a molecular net in 𝐿–ts (𝐿𝑋, 𝜏), 𝜇 ∈ 𝐿𝑋. Then:

(i) 𝑁𝛼𝐵. lim(𝑆) = 𝑁𝛼𝐵. lim(𝐼 (𝑆)).

(ii) 𝑁𝛼𝐵.adh(𝑆) ≤ 𝑁𝛼𝐵.adh(𝐼 (𝑆)).

Proof. (i) Let 𝑥𝛼 ∈ 𝑀 (𝐿𝑋) such that 𝑥𝛼 ∈ 𝑁𝛼𝐵. lim(𝑆), by Theorem 3.25 (i),

we have 𝑆
𝑁𝛼𝐵−−−−→ 𝑥𝛼 then for each 𝜆 ∈ 𝑁𝛼𝐵𝑅𝑥𝛼 there is 𝑛 ∈ 𝐷 such that for each

𝑚 ∈ 𝐷 and 𝑚 ≥ 𝑛 we have 𝑆(𝑚) ∉ 𝜆, and hence by the definition of 𝐼 (𝑆) we have



N. A. Alsaedi / Eur. J. Pure Appl. Math, 18 (4) (2025), 5960 15 of 22

𝜆 ∈ 𝐼 (𝑆) for each 𝜆 ∈ 𝑁𝛼𝐵𝑅𝑥𝛼 and so 𝑁𝛼𝐵𝑅𝑥𝛼 ⊆ 𝐼 (𝑆). Thus 𝑥𝛼 ∈ 𝑁𝛼𝐵. lim(𝐼 (𝑆)), i.e.,
𝑁𝛼𝐵. lim(𝑆) ≤ 𝑁𝛼𝐵. lim(𝐼 (𝑆)).

Conversely, let 𝑥𝛼 ∈ 𝑁𝛼𝐵. lim(𝐼 (𝑆)), then 𝜆 ∈ 𝐼 (𝑆) for each 𝜆 ∈ 𝑁𝛼𝐵𝑅𝑥𝛼 , and then
there is 𝑛 ∈ 𝐷 such that for each 𝑚 ∈ 𝐷 and 𝑚 ≥ 𝑛 we have 𝑆(𝑚) ∉ 𝜆. Therefore 𝑥𝛼 ∈
𝑁𝛼𝐵. lim(𝑆). Thus 𝑁𝛼𝐵. lim(𝐼 (𝑆)) ≤ 𝑁𝛼𝐵. lim(𝑆). Hence 𝑁𝛼𝐵. lim(𝑆) = 𝑁𝛼𝐵. lim(𝐼 (𝑆)).

(ii) Let 𝑥𝛼 ∈ 𝑁𝛼𝐵.adh(𝑆), then for each 𝜆 ∈ 𝑁𝛼𝐵𝑅𝑥𝛼 and for each 𝑛 ∈ 𝐷 such
there is 𝑚 ∈ 𝐷 and 𝑚 ≥ 𝑛 we have 𝑆(𝑚) ∉ 𝜆. If 𝜇 ∈ 𝐼 (𝑆), then there is 𝑛 ∈ 𝐷

such that for each 𝑚 ∈ 𝐷 and 𝑚 ≥ 𝑛 then 𝑆(𝑚) ∉ 𝜇. Thus 𝜇 ∨ 𝜆 ≠ 1𝑋 for each
𝜇 ∈ 𝐼 (𝑆) and for each 𝜆 ∈ 𝑁𝛼𝐵𝑅𝑥𝛼 , that is, 𝑥𝛼 ∈ 𝑁𝛼𝐵.adh(𝐼 (𝑆)). This implies that
𝑁𝛼𝐵.adh(𝑆) ≤ 𝑁𝛼𝐵.adh(𝐼 (𝑆)).

Theorem 3.27. Let (𝐿𝑋, 𝜏) be a 𝐿−ts. and let 𝜇 ∈ 𝐿𝑋. Then:

(i) If 1𝑋 is 𝑁𝑄𝛼−compact iff 1𝑋 is 𝑁𝛼−bounded.

(ii) If 𝜇 is 𝑁𝑄𝛼−compact, then 𝜇 is 𝑁𝛼−bounded.

(iii) If 𝜂 is 𝑁𝑄𝛼−compact and 𝜇 ≤ 𝜂, then 𝜇 is 𝑁𝛼−bounded.

(iv) If 𝜇1, 𝜇2, ..., 𝜇𝑚 are 𝑁𝛼−bounded sets, then
∨𝑚

𝑖=1 𝜇𝑖 is 𝑁𝛼−bounded.

Proof (i) Let (𝐿𝑋, 𝜏) be a 𝑁𝑄𝛼−compact space and let Ψ = {𝜆𝑖 : 𝑖 ∈ 𝐼} ⊆ 𝜏′ be
an 𝛼−RF of 1𝑋. Since (𝐿𝑋, 𝜏) is a 𝑁𝑄𝛼−compact space, there exists a finite subfamily
Ψ◦ = {𝜆𝑖 : 𝑖 = 1, 2, ..., 𝑚} ∈ 2(Ψ) such that Ψ◦ is an 𝛼−RCRF of 1𝑋 and so 1𝑋 is
𝑁𝛼−bounded set. Conversely, let 1𝑋 be a 𝑁𝛼−bounded set and let Ψ = {𝜆𝑖 : 𝑖 ∈ 𝐼} ⊆ 𝜏′

be an 𝛼−RF of 1𝑋. Since 1𝑋 is a 𝑁𝛼−bounded set, then there exists a finite subfamily
Ψ◦ = {𝜆𝑖 : 𝑖 = 1, 2, ..., 𝑚} ∈ 2(Ψ) such that Ψ◦ is an 𝛼−RCRF of 1𝑋 and so 1𝑋 is a
𝑁𝑄𝛼−compact set.

(ii) Let 𝜇 be a 𝑁𝑄𝛼−compact and let Ψ = {𝜆𝑖 : 𝑖 ∈ 𝐼} ⊆ 𝜏′ be an 𝛼−RF of 1𝑋
and so Ψ is an 𝛼−RF of 𝜇. Since 𝜇 is a 𝑁𝑄𝛼−compact set, then there exists a finite
subfamily Ψ◦ = {𝜆𝑖 : 𝑖 = 1, 2, ..., 𝑚} ∈ 2(Ψ) such that Ψ◦ is an 𝛼−RCRF of 𝜇 and so 𝜇 is a
𝑁𝛼−bounded set.

(iii) Let 𝜂 be a 𝑁𝑄𝛼−compact set and 𝜇 ≤ 𝜂. Let Ψ = {𝜆𝑖 : 𝑖 ∈ 𝐼} ⊆ 𝜏′ be an 𝛼−RF
of 1𝑋 and so Ψ is 𝛼−RF of 𝜂. Since 𝜂 is a 𝑁𝑄𝛼−compact set, then there exists a finite
subfamily Ψ◦ = {𝜆𝑖 : 𝑖 = 1, 2, ..., 𝑚} ∈ 2(Ψ) such that Ψ◦ is an 𝛼−RCRF of 𝜂, since 𝜇 ≤ 𝜂,
then Ψ◦ is an 𝛼−RCRF of 𝜇 and so 𝜇 is a 𝑁𝛼−bounded set.

(iv) Let 𝜇1, 𝜇2, ..., 𝜇𝑛 be a 𝑁𝛼−bounded set and let Ψ ⊆ 𝜏′ be an 𝛼−RF of 1𝑋. Since
𝜇1, 𝜇2, ..., 𝜇𝑚 are 𝑁𝛼−bounded sets, then there exist Ψ1

◦ ,Ψ
2
◦ , ...,Ψ

𝑛
◦ ∈ 2(Ψ) such that

Ψ1
◦ ,Ψ

2
◦ , ...,Ψ

𝑛
◦ are 𝛼−RCRF of 𝜇1, 𝜇2, ..., 𝜇𝑛, respectively, and so for each 𝑥𝛼 ∈ 𝜇1 there is

𝜆1 ∈ Ψ1
◦ such that 𝜆1 ∈ 𝑅𝑥𝛼 , for each 𝑥𝛼 ∈ 𝜇2, there is 𝜆2 ∈ Ψ2

◦ such that 𝜆2 ∈ 𝑅𝑥𝛼 , ..., for
each 𝑥𝛼 ∈ 𝜇𝑛 there is 𝜆𝑛 ∈ Ψ𝑛

◦ such that 𝜆𝑛 ∈ 𝑅𝑥𝛼 . Hence, for each 𝑥𝛼 ∈ 𝜇1 ∨ 𝜇2 ∨ ... ∨ 𝜇𝑛,
we have 𝑥𝛼 ∈ 𝜇𝑖 for some 𝑖 ∈ {1, 2, ..., 𝑛} and so there is 𝜆𝑖 ∈

∨𝑛
𝑖=1 Ψ

𝑖
◦ such that 𝜆𝑖 ∈ 𝑅𝑥𝛼

and hence
∨𝑛

𝑖=1 Ψ
𝑖
◦ is an 𝛼−RCRF of 𝜇1 ∨ 𝜇2 ∨ ... ∨ 𝜇𝑛. Thus Ψ𝑜 = ∨𝑛

𝑖=1Ψ
𝑜
𝑖
is 𝛼−RCRF of

𝜇1 ∨ 𝜇2 ∨ . . . ∨ 𝜇𝑛 and so ∨{𝜇𝑖 : 𝑖 = 1, 2, . . . , 𝑛} is a 𝑁𝛼−bounded set.
The following Example shows that the converse of Theorem 3.27 (ii) is not true in general.



N. A. Alsaedi / Eur. J. Pure Appl. Math, 18 (4) (2025), 5960 16 of 22

Example 3.28. Let 𝑋 = {𝑥}, 𝐿 = [0, 1], and let 𝜏 = {0𝑋, 𝑥 1
4
, 𝑥 8

9
, 1𝑋}. Then (𝐿𝑋, 𝜏) is

𝐿−ts and 𝜏′ = {0𝑋, 𝑥 3
4
, 𝑥 1

9
, 1𝑋}. Firstly, we show that 𝜇 = 𝑥 1

2
∈ 𝑀 (𝐿𝑋) is 𝑁𝛼–bounded

set. We suppose that 𝑆 = {𝑥𝛼 : 𝑥 ∈ 𝑋, 𝛼 ≤ 1
2 } is any constant 𝛼–molecular net in 𝜇. If

𝛼 ≤ 1
2 , we take 𝑥 ∈ 𝑋 so that 𝑆(𝑛) = 𝑥𝛼 ∉ 𝑐𝑙 (int(𝜆)) for each 𝜆 ∈ 𝑅𝑥0.5 = {0𝑋, 𝑥 1

9
}, where

𝑐𝑙 (int(0𝑋)) = 0𝑋 and 𝑐𝑙 (int(𝑥 1
9
)) = 0𝑋. Then 𝑥𝛼 is a 𝛿–cluster point of 𝑆 in 1𝑋. Thus

𝜇 = 𝑥0.5 is a 𝑁𝛼–bounded set.
Now, we show that 𝜇 = 𝑥0.5 ∈ 𝑀 (𝐿𝑋) is not a 𝑁𝑄𝛼–compact set. Indeed, (𝐿𝑋, 𝜏) is

not 𝐿𝑅2–space. Since there is 𝑥0.8 ∈ 𝑀 (𝐿𝑋) and there is 𝜆 = 𝑥 3
4
∈ 𝑅𝑥0.8 such that for

each 𝜂 ∈ 𝑅𝑥0.8 = {0𝑋, 𝑥 1
9
, 𝑥 3

4
} we have 𝜆 ≰ int(𝜂), where int(0𝑋) = 0𝑋, int(𝑥 1

9
) = 0𝑋 and

int(𝑥 3
4
) = 𝑥 1

4
. Hence (𝐿𝑋, 𝜏) is not a 𝐿𝑅2–space, and so 𝜇 = 𝑥0.5 is not a 𝑁𝑄𝛼–compact set.

Theorem 3.29: Every 𝐿−subset with finite support is a 𝑁𝛼−bounded set.
Proof. Let (𝐿𝑋, 𝜏) be a 𝐿−ts. and 𝜇 ∈ 𝐿𝑋 with finite support. Then by Theorem 2.17

(i), we have 𝜇 is 𝑁𝑄𝛼−compact and by Theorem 3.27 (ii) we have 𝜇 is a 𝑁𝛼−bounded
set.

Theorem 3.30. Let (𝐿𝑋, 𝜏) be an L-ts and 𝜇 be a 𝑁𝛼−bounded set in (𝐿𝑋, 𝜏), then
𝜇 is a 𝑁𝛼−bounded set in (𝐿𝑋, 𝜏𝑌 ).

Proof. Let 𝜇 be a 𝑁𝛼–bounded set in (𝐿𝑋, 𝜏) and 𝜙 ≠ 𝑌 ⊆ 𝑋. Let Ψ = {𝜌𝑖 = 𝜂𝑖 ∧ 1𝑌 :
𝜂𝑖 ∈ 𝜏′, 𝑖 ∈ 𝐼} ⊆ 𝜏′

𝑌
be an 𝛼–RF of 1𝑌 . Hence Ψ

∗ = {𝜂𝑖 : 𝑖 ∈ 𝐼} ⊆ 𝜏′ is an 𝛼–RF of 1𝑋. Since
𝜇 is a 𝑁𝛼–bounded set in (𝐿𝑋, 𝜏), then there exists Ψ∗

◦ = {𝜂𝑖𝑚 : 𝑚 = 1, 2, . . . , 𝑛} ∈ 2(Ψ
∗ )

such that Ψ∗
◦ is an 𝛼–RCRF of 𝜇 and so Ψ◦ = {𝜌𝑖𝑚 = 𝜂𝑖𝑚 ∧ 1𝑌 : 𝑚 = 1, 2, . . . , 𝑛} ∈ 2(Ψ) is

an 𝛼–RCRF of 𝜇. Hence 𝜇 is a 𝑁𝛼–bounded set in (𝐿𝑌 , 𝜏𝑌 ).

4. 𝛼–Nets’ characterizations of 𝑁.𝛼–Boundedness

In this section, we give several characterizations of N.𝛼–Boundedness in terms of both
the upper 𝛿–limit of nets of 𝐿–subsets and 𝛿–cluster points of constant molecular 𝛼–nets.

Theorem 4.1. Let (𝐿𝑋, 𝜏) be an L – ts, 𝛼 ∈ 𝑀 (𝐿) and 𝜇 ∈ 𝐿𝑋. Then 𝜇 is
N.𝛼–bounded iff for each constant molecular 𝛼–net 𝑆 contained in 𝜇 has 𝛿–cluster point
in 𝑋 with height 𝛼.

Proof. Suppose that 𝜇 is N.𝛼–bounded and let 𝑆 = {𝑆(𝑛) : 𝑛 ∈ 𝐷} be a constant
molecular 𝛼–net in 𝜇 with height 𝛼. If 𝑆 does not have any 𝛿–cluster point in 𝑋 with
height 𝛼. Then for all 𝑥𝛼 ∈ 𝑀 (𝐿𝑋), 𝑥𝛼 is not a 𝛿–cluster point of 𝑆 and so there exists
𝜆𝑥 ∈ 𝑅𝑥𝛼 and 𝑛𝑥 ∈ 𝐷 such that for every 𝑚 ∈ 𝐷 and 𝑚 ≥ 𝑛𝑥 , then 𝑆(𝑚) ∈ 𝑐𝑙 (𝑖𝑛𝑡 (𝜆𝑥)). Put
Ψ = {𝜆𝑥 : 𝑥 ∈ 𝑋 and 𝛼 ∈ 𝑀 (𝐿)} is an 𝛼–RF of 1𝑋. Since 𝜇 is N.𝛼–bounded, then there
exist Ψ𝑜 = {𝑐𝑙 (𝑖𝑛𝑡 (𝜆𝑥)) : 𝑖 = 1, 2, . . . , 𝑘} ∈ 2(Ψ) such that Ψ𝑜 is an 𝛼–RCRF of 𝜇. Hence,
for each 𝑖 ≤ 𝑘 we have 𝑛𝑥𝑖 ∈ 𝐷 when 𝑚 ≥ 𝑛𝑥𝑖 , 𝑆(𝑚) ∈ 𝑐𝑙 (𝑖𝑛𝑡 (𝜆𝑥𝑖 )). Since 𝐷 is a directed set,
then there is 𝑛𝑜 ∈ 𝐷 such that 𝑛𝑜 ≥ 𝑛𝑥𝑖 (𝑖 = 1, 2, . . . , 𝑘). Hence 𝑆(𝑚) ∈ ∧𝑘

𝑖=1 𝑐𝑙 (int(𝜆𝑥𝑖 ))
whenever 𝑚 ≥ 𝑛𝑜. This means that 𝑆(𝑚) not have 𝛼–RCRF in Ψ𝑜, and so Ψ𝑜 is not
𝛼–RCRF of 𝜇. This contradicts the hypothesis that 𝜇 is N.𝛼–bounded. Thus 𝑆 has a
𝛿–cluster point in 𝑋 with height 𝛼.



N. A. Alsaedi / Eur. J. Pure Appl. Math, 18 (4) (2025), 5960 17 of 22

Conversely, suppose that 𝜇 is not N.𝛼–bounded. Then there exist 𝛼 ∈ 𝑀 (𝐿) and
a family Ψ which is an 𝛼–RF of 1𝑋, but for any family Ψ𝑜 ∈ 2(Ψ) , we have Ψ𝑜 is not
an 𝛼–RCRF of 𝜇. Then there is a point 𝑥𝛼 ∈ 𝜇 with height 𝛼 such that 𝑥𝛼 ≤ ∧Ψ𝑜

and 𝑥𝛼 is denoted by (𝑥(Ψ𝑜))𝛼. Since 2(Ψ) is a directed set with relation ≤, then
𝑆 = {𝑥((Ψ𝑜))𝛼 : Ψ𝑜 ∈ 2(Ψ) } is a constant molecular 𝛼–net in 𝜇. Take an arbitrary point
𝑦𝛼 in 𝑋 with height 𝛼, since Ψ is an 𝛼–RF of 1𝑋, then there is 𝜆 ∈ Ψ such that 𝜆 ∈ 𝑅𝑦𝛼 .
Hence, for each Ψ𝑜 ∈ 2(Ψ) such that 𝑐𝑙 (𝑖𝑛𝑡 (𝜆)) ∈ Ψ𝑜, there is (𝑥(Ψ𝑜))𝛼 ≤ ∧Ψ𝑜 ≤ 𝑐𝑙 (𝑖𝑛𝑡 (𝜆)),
i.e., 𝑆 ≤ 𝑐𝑙 (𝑖𝑛𝑡 (𝜆)). This shows that 𝑦𝛼 is not a 𝛿–cluster point of 𝑆 in 𝑋 with height 𝛼,
which contradicts the hypothesis. Thus 𝜇 is a N.𝛼–bounded.

Theorem 4.2. (Alexander’s subbase lemma) Suppose that (𝐿𝑋, 𝜏) is a L – ts,
𝛼 ∈ 𝑀 (𝐿), 𝜇 ∈ 𝐿𝑋 and 𝜉 is a subbase of 𝜏′. If for each 𝛼–RF Ψ of 1𝑋 consisting of
elements of 𝜉, there is Ψ𝑜 ∈ 2(Ψ) which is an 𝛼–RCRF of 𝜇, then 𝜇 is a N.𝛼–bounded set.

Proof. It is similar to Theorem 5.1 in [15].

Theorem 4.3. Let {(𝐿𝑋𝑖 , 𝜏𝑖) : 𝑖 ∈ 𝐼} be a 𝐿–ts’s and 𝜇𝑖 be a 𝑁𝛼–bounded set in
(𝐿𝑋𝑖 , 𝜏𝑖) for each 𝑖 ∈ 𝐼, then the product set 𝜇 =

∏
𝑖∈𝐼 𝜇𝑖 is a 𝑁𝛼–bounded in the product

space.
Proof. Let 𝛼 ∈ 𝑀 (𝐿) and let 𝜇𝑖 ∈ 𝐿𝑋𝑖 be a 𝑁𝛼–bounded set in (𝐿𝑋𝑖 , 𝜏𝑖) for each

𝑖 ∈ 𝐼. Let the set 𝜉 = {𝑃−1
𝑖

(𝜆𝑖) : 𝑖 ∈ 𝐼, 𝜆𝑖 ∈ 𝜏′
𝑖
} be a subbase of the family of all closed sets

of the product space. To show that
∏

𝑖∈𝐼 𝜇𝑖 is a 𝑁𝛼–bounded set, we only need to verify
that for each 𝛼–RF, Ψ ⊆ 𝜉 of the set

∏
𝑖∈𝐼 𝑋𝑖 there exists Ψ𝑜 ∈ 2(Ψ) such that Ψ𝑜 is an

𝛼–RCRF of the set
∏

𝑖∈𝐼 𝜇𝑖. Let Ψ = ∨𝑛∈N{𝑃−1
𝑖𝑛

(𝜆) : 𝜆 ∈ ℜ𝑖𝑛 ,ℜ𝑖𝑛 ⊆ 𝜏′
𝑖𝑛
}.

Now, we consider the following two cases:
(i) There exists 𝑖𝑜 ∈ 𝐼 such that no molecular with height 𝛼 is contained in 𝜇𝑖𝑜 .

Then by the definition of a product set
∏

𝑖∈𝐼 𝜇𝑖 it follows immediately that no point no
molecular with height 𝛼 is contained in

∏
𝑖∈𝐼 𝜇𝑖 and hence for each Ψ𝑜 ∈ 2(Ψ) , we have

Ψ𝑜 is an 𝛼–RCRF of
∏

𝑖∈𝐼 𝜇𝑖 and so
∏

𝑖∈𝐼 𝜇𝑖 is a 𝑁𝛼–bounded set.
(ii) For every 𝑖 ∈ 𝐼, 𝑋𝑖 contains a molecular with height 𝛼, 𝑥𝑖𝛼 say. Then there

must be some 𝑛 ∈ N such that ℜ𝑖𝑛 is an 𝛼–RF of 𝑋𝑖𝑛 . In fact, for each 𝑛 ∈ N, ℜ𝑖𝑛

is not an 𝛼–RF of 𝑋𝑖𝑛 , then there exists 𝑦𝑖𝑛 ∈ 𝑋𝑖𝑛 with 𝑦
𝑖𝑛
𝛼 ∈ 1𝑋𝑖𝑛

∧ (∧ℜ𝑖𝑛). For each
𝑖 ∉ {𝑖𝑛 : 𝑛 ∈ N} take 𝑦𝑖 = 𝑥𝑖. Let 𝑦 ∈ ∏

𝑖∈𝐼 𝑋𝑖 with projections 𝑦𝑖 and 𝑖 ∈ 𝐼. On the
other hand, for each 𝜂 ∈ Ψ, 𝑦𝛼 ∈ 𝜂. To see this, let 𝜂 = 𝑃−1

𝑖𝑛
(𝜆), where 𝜆 ∈ ℜ𝑖𝑛 , then

𝜂(𝑦) = 𝑃−1
𝑖𝑛

(𝜆) (𝑦) = 𝜆(𝑃𝑖𝑛 (𝑦)) = 𝜆(𝑦𝑖𝑛) ≥ 𝛼. Since 𝑥
𝑖𝑛
𝛼 ∈ 𝜆 and 𝑦 = (𝑦𝑖1 , 𝑦𝑖2 , ..., 𝑦𝑖𝑛), hence

𝑦𝛼 ∈ 𝜂. However, this is impossible since Ψ is an 𝛼–RF of
∏

𝑖∈𝐼 𝑋𝑖. Suppose that ℜ𝑖𝑛

is an 𝛼–RF of
∏

𝑖∈𝐼 𝑋𝑖. By the 𝑁𝛼–boundedness of 𝜇𝑖𝑛 , there exists ℜ∗
𝑖𝑛

∈ 2(ℜ𝑖𝑛 ) such

that ℜ∗
𝑖𝑛

is an 𝛼–RCRF of 𝜇𝑖𝑛 . Consider the Ψ𝑜 = {𝑃−1
𝑖𝑛

(𝑐𝑙 (𝑖𝑛𝑡 (𝜆))) : 𝜆 ∈ ℜ∗
𝑖𝑛
} which

is a finite subset of Ψ, i.e., Ψ𝑜 ∈ 2(Ψ) , if 𝑥𝛼 ∈ ∏
𝑖∈𝐼 𝜇𝑖, then 𝜇𝑖𝑛 (𝑃𝑖𝑛 (𝑥)) = 𝜇𝑖𝑛 (𝑥𝑖𝑛) ≥ 𝛼

so there is 𝜆 ∈ ℜ∗
𝑖𝑛

with 𝜆 ∈ 𝑅
𝑥
𝑖𝑛
𝛼
, i.e., 𝑐𝑙 (𝑖𝑛𝑡 (𝜆(𝑥𝑖𝑛))) = 𝑐𝑙 (𝑖𝑛𝑡 (𝜆(𝑃𝑖𝑛 (𝑥)))) ≱ 𝛼, then

𝑃−1
𝑖𝑛

(𝑐𝑙 (𝑖𝑛𝑡 (𝜆))) (𝑥) ≱ 𝛼 and therefore 𝑃−1
𝑖𝑛

(𝑐𝑙 (𝑖𝑛𝑡 (𝜆))) ∈ 𝑅𝑥𝛼 . This shows that Ψ𝑜 ∈ 2(Ψ)

is an 𝛼–RCRF of
∏

𝑖∈𝐼 𝜇𝑖 and so
∏

𝑖∈𝐼 𝜇𝑖 is a 𝑁𝛼–bounded set.

Theorem 4.4. Let 𝑆 = {𝑆(𝑛) : 𝑛 ∈ 𝐷} and 𝑇 = {𝑇 (𝑛) : 𝑛 ∈ 𝐷} be a molecular nets in
a 𝐿–ts (𝐿𝑋, 𝜏) such that 𝑇 (𝑛) ≥ 𝑆(𝑛) for each 𝑛 ∈ 𝐷 and 𝑥𝛼 ∈ 𝑀 (𝐿𝑋). Then the following
results are true:



N. A. Alsaedi / Eur. J. Pure Appl. Math, 18 (4) (2025), 5960 18 of 22

(i) If 𝑆
𝑁𝛼𝐵−−−−→ 𝑥𝛼, then 𝑇

𝑁𝛼𝐵−−−−→ 𝑥𝛼.

(ii) If 𝑆
𝑁𝛼𝐵∝ 𝑥𝛼, then 𝑇

𝑁𝛼𝐵∝ 𝑥𝛼.

Proof. (i) Let 𝑥𝛼 ∈ 𝑀 (𝐿𝑋) such that 𝑆
𝑁𝛼𝐵−−−−→ 𝑥𝛼, then for each 𝜆 ∈ 𝑁𝛼𝐵𝑅𝑥𝛼 , there

exists 𝑛 ∈ 𝐷 such that for each 𝑚 ∈ 𝐷 and 𝑚 ≥ 𝑛 then 𝑆(𝑚) ∉ 𝜆. Since 𝑇 (𝑛) ≥ 𝑆(𝑛) > 𝜆,
and so for each 𝜆 ∈ 𝑁𝛼𝐵𝑅𝑥𝛼 there exists 𝑛 ∈ 𝐷 such that for each 𝑚 ∈ 𝐷 and 𝑚 ≥ 𝑛 then
𝑇 (𝑚) ∉ 𝜆. This shows that 𝑇 is 𝑁𝛼𝐵–converges to 𝑥𝛼.

(ii) Let 𝑥𝛼 ∈ 𝑀 (𝐿𝑋) such that 𝑆
𝑁𝛼𝐵∝ 𝑥𝛼, then for each 𝜆 ∈ 𝑁𝛼𝐵𝑅𝑥𝛼 and each 𝑛 ∈ 𝐷

there exists 𝑚 ∈ 𝐷 such that 𝑚 ≥ 𝑛 then 𝑆(𝑚) ∉ 𝜆. Since 𝑇 (𝑛) ≥ 𝑆(𝑛) for each 𝑛 ∈ 𝐷,
then 𝑇 (𝑛) ≥ 𝑆(𝑛) > 𝜆. Thus for each 𝜆 ∈ 𝑁𝛼𝐵𝑅𝑥𝛼 and for each 𝑛 ∈ 𝐷 there exists 𝑚 ∈ 𝐷

such that 𝑚 ≥ 𝑛 then 𝑇 (𝑚) ∉ 𝜆. This shows that 𝑇
𝑁𝛼𝐵∝ 𝑥𝛼.

Theorem 4.5. Let {𝜇𝑛 : 𝑛 ∈ 𝐷} be a net of closed 𝐿-subsets in 𝐿𝑋 such that
𝜇𝑛1 ≤ 𝜇𝑛2 then 𝛿.lim(𝜇𝑛) ≤ ∧{𝜇𝑛 : 𝑛 ∈ 𝐷} iff 𝑛2 ≤ 𝑛1.

Proof. Let 𝑥𝛼 ∈ 𝛿.lim(𝜇𝑛) and let 𝑥𝛼 ∉ ∧{𝜇𝑛 : 𝑛 ∈ 𝐷} hence there is 𝑛◦ ∈ 𝐷 such
that 𝑥𝛼 ∉ 𝜇𝑛◦ . Let 𝜌 = 𝜇𝑛◦ , then 𝜌 ∈ 𝑅𝑥𝛼 . Since 𝑥𝛼 ∈ 𝛿.lim(𝜇𝑛) and 𝜌 ∈ 𝑅𝑥𝛼 , there
is 𝑛 ∈ 𝐷, 𝑛 ≥ 𝑛◦ such that 𝜇𝑛 ≰ 𝑐𝑙 (int(𝜌)), which contradicts the hypothesis that
𝑥𝛼 ∉ ∧{𝜇𝑛 : 𝑛 ∈ 𝐷}. Thus 𝑥𝛼 ∈ ∧{𝜇𝑛 : 𝑛 ∈ 𝐷}.

Theorem 4.6. Let (𝐿𝑋, 𝜏) be an L–ts and 𝜇 ∈ 𝐿𝑋. Then 𝜇 is a N.𝛼–bounded set iff
for every net {𝜂𝑛 : 𝑛 ∈ 𝐷} of closed 𝐿–subsets in 𝐿𝑋 such that 𝛿 lim(𝜂𝑛) (𝑥) < 𝛼, for each
𝑥 ∈ 𝑋, there is 𝑛𝑜 ∈ 𝐷 for which 𝜂𝑛 ∧ 𝜇 = 0𝑋 for every 𝑛 ∈ 𝐷, 𝑛 ≥ 𝑛𝑜.

Proof. Let 𝜇 ∈ 𝐿𝑋 be a N.𝛼–bounded set and let {𝜂𝑛 : 𝑛 ∈ 𝐷} be a net of
closed 𝐿–subsets in 𝐿𝑋 such that 𝛿.lim(𝜂𝑛) (𝑥) < 𝛼, for each 𝑥 ∈ 𝑋. Then for every
molecular 𝑥𝛼 ∈ 𝑀 (𝐿𝑋) there exists 𝜌𝑥 ∈ 𝑅𝑥𝛼 and 𝑛𝑥 ∈ 𝐷 such that 𝜂𝑛 ≤ 𝑐𝑙 (𝑖𝑛𝑡 (𝜌𝑥))
for every 𝑛 ∈ 𝐷, 𝑛 ≥ 𝑛𝑥. Since 𝜌𝑥 ∈ 𝑅𝑥𝛼 for every 𝑥 ∈ 𝑋, then the family Ψ =

{𝜌𝑥 : 𝑥 ∈ 𝑋 𝑎𝑛𝑑 𝛼 ∈ 𝑀 (𝐿)} is an 𝛼–RF of 1𝑋. Since 𝜇 is a N.𝛼–bounded, there
exist Ψ𝑜 = {𝑐𝑙 (𝑖𝑛𝑡 (𝜌𝑥𝑖 )) : 𝑖 = 1, 2, ..., 𝑘} ∈ 2(Ψ) such that Ψ𝑜 is an 𝛼–RCRF of 𝜇.
Put 𝜌 =

∧𝑘
𝑖=1 𝜌𝑥𝑖 , then 𝑐𝑙 (𝑖𝑛𝑡 (𝜌)) ∈ 𝑅𝑥𝛼 . Since 𝐷 is a directed set, there is 𝑛𝑜 ∈ 𝐷

such that 𝑛𝑜 ≥ 𝑛𝑥𝑖 for every 𝑖 = 1, 2, ..., 𝑘. Then for every 𝑛 ∈ 𝐷, 𝑛 ≥ 𝑛𝑜, we have
𝜂𝑛 ≤ 𝑐𝑙 (𝑖𝑛𝑡 (∧𝑘

𝑖=1 𝜌𝑥𝑖 )) and so 𝜂𝑛 ≤ 𝑐𝑙 (𝑖𝑛𝑡 (𝜌)), whenever 𝑛 ≥ 𝑛𝑜. Since 𝑐𝑙 (𝑖𝑛𝑡 (𝜌)) ∧ 𝜇 = 0𝑋,
then 𝜂𝑛 ∧ 𝜇 = 0𝑋 for every 𝑛 ∈ 𝐷, 𝑛 ≥ 𝑛𝑜.

Conversely, suppose that 𝜇 ∈ 𝐿𝑋 satisfies the condition of the Theorem. We prove
that 𝜇 is a N.𝛼–bounded set. Let Ψ ⊆ 𝜏′ be an 𝛼–RF of 1𝑋. Let 𝐷 = 2(Ψ) be the
set of all finite subsets of Ψ directed by inclusion, and let {𝜂Ψ : Ψ ∈ 𝐷} be a net of
closed 𝐿–subsets in 𝐿𝑋 such that 𝜂Ψ = ∧{𝑐𝑙 (𝑖𝑛𝑡 (𝜌)) : 𝜌 ∈ Ψ}. Obviously, 𝜂Ψ1 ≤ 𝜂Ψ2

iff Ψ2 ⊆ Ψ1. Hence by Theorem 4.5 it follows that 𝛿.lim(𝜂Ψ) (𝑥) ≤ ∧{𝜂Ψ : Ψ ∈ 𝐷}.
Then ∧{𝜂Ψ : Ψ ∈ 𝐷}(𝑥) = ∧(∧{𝑐𝑙 (𝑖𝑛𝑡 (𝜌)) : 𝜌 ∈ Ψ})(𝑥) < 𝛼 for each 𝑥 ∈ 𝑋. Thus
𝛿.lim(𝜂Ψ) (𝑥) < 𝛼 for each 𝑥 ∈ 𝑋. By assumption, there exists an element Ψ𝑜 ∈ 𝐷 for
which 𝜂Ψ ∧ 𝜇 = 0𝑋 for every Ψ ∈ 𝐷, Ψ ≥ Ψ𝑜. By the above we have 𝜂Ψ𝑜

∧ 𝜇 = 0𝑋 and
so (∀𝑥𝛼 ∈ 𝜇) (∃𝑐𝑙 (𝑖𝑛𝑡 (𝜌)) ∈ Ψ𝑜) (𝑐𝑙 (𝑖𝑛𝑡 (𝜌)) ∈ 𝑅𝑥𝛼). Thus Ψ𝑜 ∈ 2(Ψ) is an 𝛼–RCRF of 𝜇.
Hence 𝜇 is a N.𝛼–bounded.

Theorem 4.7. An L–ts (𝐿𝑋, 𝜏) is a 𝑁𝑄𝛼–compact iff for every a net {𝜂𝑛 : 𝑛 ∈ 𝐷}
of closed 𝐿–subsets in 𝐿𝑋 such that 𝛿lim(𝜂𝑛) (𝑥) < 𝛼 for each 𝑥 ∈ 𝑋, there is 𝑛𝑜 ∈ 𝐷 for



N. A. Alsaedi / Eur. J. Pure Appl. Math, 18 (4) (2025), 5960 19 of 22

which 𝜂𝑛 = 0𝑋 for every 𝑛 ∈ 𝐷, 𝑛 ≥ 𝑛𝑜.
Proof. Let (𝐿𝑋, 𝜏) be a 𝑁𝑄𝛼–compact. Then 1𝑋 is a 𝑁𝑄𝛼–compact and by Theorem

3.27 (i), we have 1𝑋 is a 𝑁.𝛼–bounded. Let {𝜂𝑛 : 𝑛 ∈ 𝐷} be a net of closed 𝐿–subsets in
𝐿𝑋 such that 𝛿.lim(𝜂𝑛) (𝑥) < 𝛼 for each 𝑥 ∈ 𝑋. Hence by Theorem 4.6, there exists 𝑛◦ ∈ 𝐷

such that 𝜂𝑛 ∧ 1𝑋 = 0𝑋 for every 𝑛 ∈ 𝐷, 𝑛 ≥ 𝑛◦ and so 𝜂𝑛 = 0𝑋 for every 𝑛 ∈ 𝐷, 𝑛 ≥ 𝑛◦.
Conversely, suppose that 1𝑋 satisfies the condition. We prove that 1𝑋 is a 𝑁𝑄𝛼–

compact. Let Ψ be an 𝛼–RF of 1𝑋 and let 𝐷 = 2(Ψ) be the set of all finite subsets of Ψ
directed by inclusion, and let {𝜂Ψ : Ψ ∈ 𝐷} be a net of closed 𝐿–subsets in 𝐿𝑋 such that
𝜂Ψ = ∧{𝑐𝑙 (int(𝜌)) : 𝜌 ∈ Ψ}.

Obviously, 𝜂Ψ1 ≤ 𝜂Ψ2 iff Ψ2 ⊆ Ψ1. Hence, by Theorem 4.6, it follows that

𝛿.lim(𝜂Ψ) ≤ ∧{𝜂Ψ : Ψ ∈ 𝐷}.

Then ∧{𝜂Ψ : Ψ ∈ 𝐷}(𝑥) = ∧{𝑐𝑙 (int(𝜌)) : 𝜌 ∈ Ψ}(𝑥) < 𝛼 for every 𝑥 ∈ 𝑋.and so
𝛿.lim(𝜂𝑛) (𝑥) < 𝛼, for each 𝑥 ∈ 𝑋. By assumption, there exists an element Ψ◦ ∈ 𝐷 for
which 𝜂Ψ = 0𝑋 for every Ψ ∈ 𝐷, Ψ ≥ Ψ◦. Thus 𝜂Ψ◦ = 0𝑋 and so 𝑥𝛼 ∉ 𝜂Ψ◦ = ∧{𝑐𝑙 (int(𝜌)) :
𝜌 ∈ Ψ◦} for every 𝑥𝛼 ∈ 𝑀 (𝐿𝑋) and hence Ψ◦ ∈ 2(Ψ) is an 𝛼–RCRF of 1𝑋.Hence 1𝑋 is a
𝑁𝑄𝛼–compact.

Theorem 4.8. If (𝐿𝑋, 𝜏) is fully stratified and 𝐿𝑇2–space, then 𝜇 ∈ 𝐿𝑋 is a
𝑁𝑄𝛼–compact set iff 𝜇 is 𝛿–closed and N.𝛼–bounded.

Proof. Let 𝜇 ∈ 𝐿𝑋 be an 𝑁𝑄𝛼–compact, then by Theorem 2.18 (ii), we have 𝜇 is
𝛿–closed and by Theorem 3.27 (ii), we have 𝜇 is N.𝛼–bounded. Conversely, let 𝜇 be a
𝛿–closed and N.𝛼–bounded set and let 𝑆 be a constant 𝛼–molecular net in 𝜇. Since 𝜇 is
N.𝛼–bounded, then by Theorem 4.1, we have 𝑆 has 𝛿–cluster point, say 𝑥𝛼 in 𝑋 with
height 𝛼. By Theorem 2.15 (j), then there is a subnet 𝑇 of 𝑆 such that 𝑇 𝛿–converges to
𝑥𝛼 and so 𝑥𝛼 ∈ 𝛿𝑐𝑙 (𝜇) by Theorem 3.22 (vi). Since 𝜇 is 𝛿–closed, then 𝛿𝑐𝑙 (𝜇) = 𝜇 and so
𝑥𝛼 ∈ 𝜇, then by Theorem 2.19, we have 𝜇 is a compact set.

Theorem 4.9. If (𝐿𝑋, 𝜏) is 𝐿𝑅2–space, then 𝜇 ∈ 𝐿𝑋 is a N.𝛼–bounded set iff 𝛿𝑐𝑙 (𝜇)
is a N.𝛼–bounded set.

Proof. If 𝛿.𝑐𝑙 (𝜇) is a 𝑁.𝛼–bounded set, then 𝜇 is a 𝑁.𝛼–bounded set (by Theorem
3.7). Conversely, suppose that 𝜇 is 𝑁.𝛼–bounded and Ψ = {𝜂𝑥 𝑗

: 𝑗 ∈ 𝐽} is an 𝛼–RF
of 1𝑋. Then for each 𝑥 ∈ 𝑋 there is 𝜂𝑥 𝑗

∈ Ψ such that 𝜂𝑥 𝑗
∈ 𝑅𝑥𝛼 . Since (𝐿𝑋, 𝜏) is

𝐿𝑅2–space, then there is 𝜆𝑥 𝑗
∈ 𝑅𝑥𝛼 and there is 𝜌𝑥 𝑗

∈ 𝜏′ such that 𝜆𝑥 𝑗
∨ 𝜌𝑥 𝑗

= 1𝑋 and
𝜌𝑥 𝑗

∧ 𝜂𝑥 𝑗
= 0𝑋. Then the family {𝜆𝑥 𝑗

: 𝑥𝛼 ∈ 𝑀 (𝐿𝑋)} is an 𝛼–RF of 1𝑋. Since 𝜇 is
𝑁.𝛼–bounded, then exists finite subset 𝐽◦ of 𝐽 such that {𝜆𝑥 𝑗

: 𝑗 ∈ 𝐽◦} is an 𝛼–RCRF of
𝜇. Since 𝜆𝑥 𝑗

∨ 𝜌𝑥 𝑗
= 1𝑋, 𝑥𝛼 ∉ 𝜆𝑥 𝑗

, then 𝑥𝛼 ∈ 𝜌𝑥 𝑗
. Since 𝜌𝑥 𝑗

∧ 𝜂𝑥 𝑗
= 0𝑋, then {𝜂𝑥 𝑗

: 𝑗 ∈ 𝐽◦}
is an 𝛼–RCRF of 𝜌𝑥 𝑗

. Therefore 𝜇 ≤ 𝜌𝑥 𝑗
for 𝐽 ∈ 𝐽◦. Since 𝜌𝑥 𝑗

∈ 𝜏′ and (𝐿𝑋, 𝜏) is
𝐿𝑆𝑅2–space, then by Theorem 2.21, we have 𝛿.𝑐𝑙 (𝜌𝑥 𝑗

) = 𝜌𝑥 𝑗
and so {𝜂𝑥 𝑗

: 𝑗 ∈ 𝐽◦} is an
𝛼–RCRF of 𝛿.𝑐𝑙 (𝜌𝑥 𝑗

) and since 𝛿.𝑐𝑙 (𝜇) ≤ 𝛿.𝑐𝑙 (𝜌𝑥 𝑗
), then {𝜂𝑥 𝑗

: 𝑗 ∈ 𝐽◦} is an 𝛼–RCRF of
𝛿.𝑐𝑙 (𝜇). Hence 𝛿.𝑐𝑙 (𝜇) is a 𝑁.𝛼–bounded set.

Theorem 4.10. If (𝐿𝑋, 𝜏) is a 𝐿𝑇3–space, then 𝜇 ∈ 𝐿𝑋 is a N.𝛼–bounded set iff 𝜇 is
a 𝐿–subset of a 𝑁𝑄𝛼–compact set.



N. A. Alsaedi / Eur. J. Pure Appl. Math, 18 (4) (2025), 5960 20 of 22

Proof. If 𝜇 is N.𝛼–bounded, then by Theorem 4.9, and Corollary 2.22, we have
𝛿𝑐𝑙 (𝜇) is 𝛿–closed and N.𝛼–bounded set, hence by Theorem 4.8, we have 𝛿𝑐𝑙 (𝜇) is a
𝑁𝑄𝛼–compact set.

Conversely, If 𝜇 is a 𝐿–subset of 𝑁𝑄𝛼–compact set, then by Theorem 3.27 (iii), we
have 𝜇 is a N.𝛼–bounded set.

Theorem 4.11. Assume that 𝑆 = {𝑆(𝑛) : 𝑛 ∈ 𝐷} is a molecular net in a L–ts (𝐿𝑋, 𝜏)
and 𝑥𝛼 ∈ 𝑀 (𝐿𝑋). Then the following results are true:

(i) 𝑥𝛼 is a 𝑁𝛼𝐵–cluster point of 𝑆 iff there exists a subnet 𝑇 of 𝑆 such that 𝑇 is
𝑁𝛼𝐵–converges to 𝑥𝛼.

(ii) If 𝑥𝛼 is a 𝑁𝛼𝐵–cluster point of 𝑆, then 𝑇 is a 𝑁𝛼𝐵–converges to 𝑥𝛼 for each subnet
𝑇 of 𝑆.

Proof. (i) Provided that 𝑆 = {𝑆(𝑛) : 𝑛 ∈ 𝐷} and 𝑥𝛼 is a 𝑁𝛼𝐵–cluster point of
𝑆, then for each 𝜆 ∈ 𝑁𝛼𝐵𝑅𝑥𝛼 and each 𝑛 ∈ 𝐷 there is 𝑘 ∈ 𝐷 such that 𝑆(𝑘) ∉ 𝜆 and
𝑘 ≥ 𝑛. Taking 𝑘 = 𝑔(𝑛, 𝜆), we get a mapping 𝑔 : 𝐷 × 𝑁𝛼𝐵𝑅𝑥𝛼 → 𝐷 with 𝑆(𝑔(𝑛, 𝜆)) ∉ 𝜆.
Put 𝐸 = 𝐷 × 𝑁𝛼𝐵𝑅𝑥𝛼 and we define the relation ≤ on 𝐸 as follows: (𝑛, 𝜆1) ≤ (𝑛2, 𝜆2)
iff 𝑛1 ≤ 𝑛2 and 𝜆1 ≤ 𝜆2, then (𝐸, ≤) is a directed set. For each (𝑛, 𝜆) ∈ 𝐸, we choose
𝑇 (𝑛, 𝜆) = 𝑆(𝑔(𝑛, 𝜆)), then 𝑇 = {𝑇 (𝑛, 𝜆) : (𝑛, 𝜆) ∈ 𝐸} is a subnet of 𝑆. Because: (*) There
exists mapping 𝑓 : 𝐸 → 𝐷 define as follows 𝑓 (𝑛, 𝜆) = 𝑛 and 𝑇 = 𝑆 ◦ 𝑓 .

(**) Let 𝑛1 ∈ 𝐷, then there exists (𝑛1, 𝜆1) ∈ 𝐸 and (𝑛1, 𝜆1) ≤ (𝑛2, 𝜆2) ∈ 𝐸 iff 𝑛1 ≤ 𝑛2
and 𝜆1 ≤ 𝜆2, 𝑓 (𝑛2, 𝜆2) = 𝑛2 ≥ 𝑛1. Now we prove that 𝑇 is 𝑁𝛼𝐵–converges to 𝑥𝛼, let
𝜆 ∈ 𝑁𝛼𝐵𝑅𝑥𝛼 and 𝑛 ∈ 𝐷, so (𝑛, 𝜆) ∈ 𝐸. Therefore, for each (𝑛, 𝜆) ∈ 𝐸 and (𝑛, 𝜆) ≤ (𝑚, 𝜂)
then 𝑇 (𝑚, 𝜂) = 𝑆(𝑔(𝑚, 𝜂)) ∉ 𝜂 and 𝜆 ≤ 𝜂, so 𝑇 (𝑚, 𝜂) ∉ 𝜆. Thus 𝑇 is 𝑁𝛼𝐵–converges to 𝑥𝛼.

Conversely, it follows directly from Definition 2.12.
(ii) It follows directly from Definition 2.12.

Theorem 4.12. Let (𝐿𝑋, 𝜏) be an L–ts and 𝜇 ∈ 𝐿𝑋. Then 𝜇 is a N.𝛼–bounded set
iff every 𝛼–filter F containing 𝜇 as an element has a 𝛿–cluster point in 𝑋 with height 𝛼.

Proof. Suppose that 𝜇 is a N.𝛼–bounded and F is a 𝛼–filter containing 𝜇 as an
element (𝛼 ∈ 𝑀 (𝐿)), then 𝜆 ∧ 𝜇 ∈ F for each 𝜆 ∈ F , hence

∨
𝑥∈𝑋

(𝜆 ∧ 𝜇) (𝑥) ≥ 𝛼 for each

𝜆 ∈ F and for each 𝑥𝛼 ∈ 𝑀 (𝐿𝑋) there exists a molecule 𝑥 (𝜆,𝛼) ∈ 𝜆 ∧ 𝜇 with height 𝛼.
Put 𝑆(F ) = {𝑥 (𝜆,𝛼) : (𝜆, 𝛼) ∈ F × 𝑀 (𝐿)}. In F × 𝑀 (𝐿) we define the relation that
(𝜆1, 𝛼1) ≥ (𝜆2, 𝛼2)𝑖 𝑓 𝑓 𝜆1 ≤ 𝜆2 and 𝛼1 ≥ 𝛼2. Then, F × 𝑀 (𝐿) is a directed set with this
relation, and 𝑆(F ) is a constant molecular 𝛼–net in 𝜇. Since 𝜇 is a N.𝛼–bounded set,
then by Theorem 4.1, 𝑆(F ) has a 𝛿–cluster point in 𝑋 with height 𝛼, say 𝑥𝛼. So, by
Theorem 2.28, F 𝛿–cluster to 𝑥𝛼 as well.

Conversely, suppose that the condition is satisfied and 𝑆 = {𝑆(𝑛) : 𝑛 ∈ 𝐷} is a constant
molecular 𝛼–net in 𝜇. Let 𝜆𝑚 =

∨(𝑆(𝑚)) for each 𝑚 ∈ 𝐷, 𝑛 ≥ 𝑚. Since 𝐷 is a directed
set, then the family {𝜆𝑚 : 𝑚 ∈ 𝐷} can generate a filter F (𝑆). Since 𝑆 is a constant
molecular 𝛼–net, then for each 𝛼 ∈ 𝑀 (𝐿) (∃𝑛 ∈ 𝐷) (∀𝑚 ∈ 𝐷, 𝑚 ≥ 𝑛) (∨(𝑆(𝑚)) = 𝛼), hence∨(𝜆𝑚(𝑥)) =

∨(∨(𝑆(𝑛))) = 𝛼, 𝑛 ≥ 𝑚 and so
∨(𝜆𝑚(𝑥)) = 𝛼. Since F (𝑆) is produced by

{𝜆𝑚 : 𝑚 ∈ 𝐷}, then for each 𝜆 ∈ F (𝑆) contains some 𝜆𝑚 and therefore
∨(𝜆(𝑥)) = 𝛼.

Hence F (𝑆) is an 𝛼–filter. By assumption, F (𝑆) has a 𝛿–cluster point in 𝑋 with height
𝛼, say 𝑥𝛼. Thus, for each 𝜇 ∈ 𝑅𝑥𝛼 and for each 𝜆 ∈ F (𝑆). In particular, 𝜆𝑚 we have



N. A. Alsaedi / Eur. J. Pure Appl. Math, 18 (4) (2025), 5960 21 of 22

𝜆𝑚 ≰ 𝜇, and by Theorem 2.29, we have 𝑆 has a 𝛿–cluster point 𝑥𝛼 and by Theorem 4.1,
we have 𝜇 is a N.𝛼–bounded.

Theorem 4.13. If a set 𝜇 in a L–ts (𝐿𝑋, 𝜏) is a N.𝛼–bounded, then every 𝛼–ideal 𝐼
in 𝐿𝑋 and 𝜇 ∉ 𝐼 has a 𝛿–cluster point in 𝑋 with height 𝛼.

Proof. Let 𝐼 be an 𝛼–ideal in 𝐿𝑋 and 𝜇 ∈ 𝐿𝑋 be a 𝑁𝛼–bounded with 𝜇 ∉ 𝐼. Then for
each 𝜂 ∈ 𝐼 we have

∨
𝑥∈𝑋 𝜂(𝑥) < 𝛼, and then for each 𝛼 ∈ 𝑀 (𝐿) there exists a molecule

𝑆(𝜂, 𝛼) = 𝑥 (𝜂,𝛼) ∉ 𝜂. Put 𝐷 (𝐼) = {(𝜂, 𝛼) : 𝑥 (𝜂,𝛼) ∈ 𝜇, 𝜂 ∈ 𝐼 and 𝑥 (𝜂,𝛼) ∉ 𝜂}.In 𝐷 (𝐼) We
define the relation that (𝜂1, 𝛼1) ≥ (𝜂2, 𝛼2) iff 𝜂1 ≥ 𝜂2. Then (𝐷 (𝐼), ≥) is a directed set
with this relation and 𝑆(𝐼) = {𝑆(𝜂, 𝛼) = 𝑥 (𝜂,𝛼) : (𝜂, 𝛼) ∈ 𝐷 (𝐼)} is a constant molecular
𝛼–net in 𝜇. Since 𝜇 is a 𝑁𝛼–bounded set, then by Theorem 4.1, 𝑆(𝐼) has a 𝛿–cluster
point in 𝑋 with height 𝛼, say 𝑥𝛼, by Theorem 2.30, we have 𝑥𝛼 is also a 𝛿–cluster point
of 𝐼.

Acknowledgements

The author is thankful to the referees for their kind suggestions, which considerably
improved the presentation of the paper.

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