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On I-convergence of nets of functions in fuzzy metric spaces

  • Lingsheng Zhong , Wenfei Xi and Zhongbao Tang EMAIL logo
Published/Copyright: December 22, 2025
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Open Mathematics
From the journal Open Mathematics Volume 23 Issue 1

Abstract

In this paper, we introduce ideal versions of semi-α convergence, semi-exhaustiveness and semi uniform convergence of nets of functions between two fuzzy metric spaces, and obtain some properties of them. Let I be a D-admissible ideal on the directed set (D, ≥), X and Y be two fuzzy metric spaces, ( f d ) d ∈ D be a net of functions in Y X and f ∈ Y X . We mainly show that: (1) If ( f d ) d ∈ D is a net of continuous functions and I -semi uniformly convergent to f, then f is continuous; (2) The following are equivalent: (a) the net ( f d ) d ∈ D is I -semi-exhaustive and pointwise I -convergent to f; (b) the net ( f d ) d ∈ D is I -semi-α convergent to f; (c) the net ( f d ) d ∈ D is I -semi uniformly convergent to f and the function f is continuous.

Keywords and phrases: fuzzy metric spaces; I -semi-α convergence; I -semi-exhaustiveness; I -semi uniform convergence; nets of functions
MSC 2020: 40A35; 54A20; 54A40; 40A99

1 Introduction

Fuzzy sets were initially introduced by Zadeh [1]. After that, fuzzy sets have been applied in many fields of mathematics. The concept of fuzzy metric space was introduced by many authors in different ways. In particular, George and Veeramani [2] modified the concept of fuzzy metric space introduced by Kramosil and Michálek [3] with the help of continuous t-norms and defined a Hausdorff topology on this modified fuzzy metric space. Moreover, the topology induced by this fuzzy metric is first countable [2] and metrizable [4]. It is worth noticing that numerous concepts and results from classical metric spaces have been generalized to fuzzy metric spaces [2],[4], [5], [6], [7].

The idea of statistical convergence is an extension of usual convergence, which was first given by Zygmund in the first edition of his monograph [8] in 1935. In 1951, Fast [9] and Steinhaus [10] independently gave the concept of statistical convergence of real number sequences based on the asymptotic density of subsets of positive integers. Although statistical convergence was introduced over nearly the last 90 years, it has become an active area of research for 40 years with the contributions of several authors, Šalát [11], Fridy [12],13], Di Maio and Kočinac [14]. Statistical convergence has been widely applied in different fields of mathematics. See [14], [15], [16], [17], [18], [19] etc.

The idea of statistical convergence has been extended to I -convergence by Kostyrko et al. in [20] with the help of ideals. I -convergence includes usual convergence and statistical convergence when I is, respectively, the ideal of all finite subsets of the set of positive integers and the ideal of all subsets of the set of positive integers with natural density zero. Over the last 20 years, a lot of work has been done on I -convergence and associated topics, and it has turned out to be one of the most active research areas in Topology and Analysis. For more details see [21], [22], [23], [24], [25], [26], [27], [28].

It is well known that the pointwise limit of a sequence of continuous functions is not necessarily a continuous function. So one of the central questions in analysis is what precisely must be added to pointwise convergence of a sequence of continuous functions to preserve the continuity of the limit function. In 1841, Weierstrass gave a sufficient condition called uniform convergence which yields the continuity of the limit function. In 1878, Dini [29] gave another sufficient condition, weaker than uniform convergence, for continuity of the limit function. In 1883–1884, Arzelà [30] discovered a necessary and sufficient condition under which the pointwise limit of a sequence of real valued continuous functions on a compact interval is continuous. He called this condition “uniform convergence by segment”, which was called “quasi-uniform convergence” by Borel [31]. The concept of α-convergence (also named as continuous convergence) has been known at the beginning of the 20th century. Later, around 1950, Arens [32], Hahn [33], Stoilov [34] and Iséki [35] gave some characterizations of α-convergence. One of the interesting facts about α-convergence (proved by Stoilov [34]) is that it preserves the continuity of the limit function for sequences of functions in metric spaces. In 2008, Gregoriades and Papanastassiou [36] introduced the concept of exhaustiveness and established a connection between α-convergence and pointwise convergence by this concept. Based on the relevant concepts of statistical convergence, Caserta and Kocinač [16] considered statistical versions of exhaustiveness and α-convergence of sequences of functions between two metric spaces. Papachristodoulos, Papanastassiou and Wilczynski [37] introduced ideal versions of exhaustiveness and α-convergence, and Megaritis [38] studied ideal version of uniform convergence.

Recently, Papanastassiou [39] introduced the concepts of semi-α convergence, semi-exhaustiveness and semi uniform convergence of sequences of functions between metric spaces, which are strictly weaker than α-convergence, exhaustiveness and almost uniform convergence, respectively. Let (X, d) and (Y, ρ) be two metric spaces, ( f n ) n ∈ N be a sequence of functions in Y X and f ∈ Y X . The following facts are proved in [39]: (1) If ( f n ) n ∈ N is semi-α convergent to f, then f is continuous. (2) If ( f n ) n ∈ N is pointwise convergent to f and semi-exhaustive, then f is continuous. (3) For a pointwise convergence of sequence of functions ( f n ) n ∈ N , the concepts of semi-α convergence and semi-exhaustiveness are equivalent. (4) The sequence of functions ( f n ) n ∈ N is semi-α convergent to f if and only if ( f n ) n ∈ N is semi uniformly convergent to f and f is continuous.

Based on the analysis above, it is natural to consider ideal versions of semi-α convergence, semi-exhaustiveness and semi uniform convergence of nets of functions between two fuzzy metric spaces (in the sense of George and Veeramani), which is the main purpose of this paper. The outline of this paper is as follows. In Section 2, we review some necessary definitions and results. In Section 3, we define ideal versions of semi-α convergence, semi-exhaustiveness and semi uniform convergence of nets of functions, and clarify their relations with related concepts. Let I be a D-admissible ideal on the directed set (D, ≥), X and Y be two fuzzy metric spaces, ( f d ) d ∈ D be a net of functions in Y X and f ∈ Y X . We mainly show that: (1) If ( f d ) d ∈ D is a net of continuous functions and I -semi uniformly convergent to f, then f is continuous; (2) The following are equivalent: (a) the net ( f d ) d ∈ D is I -semi-exhaustive and pointwise I -convergent to f; (b) the net ( f d ) d ∈ D is I -semi-α convergent to f; (c) the net ( f d ) d ∈ D is I -semi uniformly convergent to f and the function f is continuous. These results extend the corresponding results in [39].

2 Preliminaries

Throughout this paper, all fuzzy metric spaces are in the sense of George and Veeramani. N denotes the set of all positive integers with the natural ordering and Y X (resp. C(X, Y)) denotes the set of all functions (resp. all continuous functions) from X to Y. Readers may consult [40] for notation and terminology not given here.

We begin with some basic concepts about the fuzzy metric spaces.

Definition 2.1.

[2] A binary operation ∗: [0, 1] × [0, 1] → [0, 1] is called a continuous t-norm if

  1. ∗ is associative and commutative;

  2. ∗ is continuous;

  3. for all a ∈ [0, 1], a∗1 = a;

  4. for all a, b, c, d ∈ [0, 1], a ≤ c, b ≤ d ⇒ a∗b ≤ c∗d.

Typical examples of continuous t-norms are a∗b = ab, a∗b = min{a, b}, and a∗b = max{0, a + b − 1}.

Definition 2.2.

[2] A triple (X, M, ∗) is a fuzzy metric space if X is a set, ∗ is a continuous t-norm, and M is a function

M : X × X × ( 0 , ∞ ) → [ 0,1 ]

satisfying the following conditions for all x, y, z ∈ X and all s, t > 0:

  1. M(x, y, t) > 0;

  2. M(x, y, t) = 1 if and only if x = y;

  3. M(x, y, t) = M(y, x, t);

  4. M(x, y, t)∗M(y, z, s) ≤ M(x, z, t + s);

  5. M(x, y, ⋅): (0, ∞) → [0, 1] is continuous.

The adjective “Fuzzy” in the above definition comes from the fact that the function M is a fuzzy set. The value of M(x, y, t) is usually understood as the degree of certainty that the distance between x and y is less than t. It was shown in [41] that M(x, y, ⋅) is non-decreasing for all x, y in X.

Definition 2.3.

[2] Let (X, d) be a metric space and a∗b = ab for any a, b ∈ [0, 1]. Define

M d ( x , y , t ) = t t + d ( x , y ) .

Then (X, M d , ∗) is fuzzy metric space. We call this fuzzy metric space the standard fuzzy metric space and M d the standard fuzzy metric induced by the metric d.

It is worth noticing that there exist fuzzy metric spaces which are not standard fuzzy metric spaces for any metric (see [2], Example 2.11]).

In this paper, R denotes the set of all real numbers, R M | ⋅ | denotes the standard fuzzy metric space ( R , M | ⋅ | , ⋅ ) where |⋅| is the Euclidean metric in the real line.

Definition 2.4.

[2] Let (X, M, ∗) be a fuzzy metric space. Then the ball B(x, ɛ, t) with center x ∈ X and ɛ ∈ (0, 1), t > 0 is defined by

B ( x , ε , t ) = { y ∈ X ∣ M ( x , y , t ) > 1 − ε } .

It was shown in [2] that the collection of all balls in a fuzzy metric space (X, M, ∗) induces a first countable and Hausdorff topology τ M on X. In fact, this topology is always metrizable (see [4]). When the fuzzy metric space is the standard fuzzy metric space induced by the metric d, the topology induced by fuzzy metric coincides with the topology induced by the metric d [2].

A sequence in a fuzzy metric space (X, M, ∗) is said to be convergent if it is convergent in (X, τ M ), where τ M is the topology induced by the fuzzy metric M [2]. The following proposition gives a characterization of the convergent sequences in fuzzy metric spaces.

Proposition 2.5.

[2] Let (X, M, ∗) be a fuzzy metric space. A sequence ( x n ) n ∈ N in X converges to x ∈ X if and only if lim n → ∞ M ( x n , x , t ) = 1 for all t > 0.

From Proposition 2.5, it is clear that ( x n ) n ∈ N converges to x if and only if for each ɛ ∈ (0, 1) and t > 0, there is n 0 ∈ N such that M(x n , x, t) > 1 − ɛ for all n ≥ n 0.

We now recall the concept of I -convergence of nets. For more details we refer the readers to [23],24].

Definition 2.6.

[40] Recall some basic concepts about nets.

  1. Let D be a non-empty set and “≥” be a binary relation on D. The pair (D, ≥) is said to be a directed set if: (a) “≥” is reflexive and transitive; (b) for any two elements d 1, d 2 ∈ D, there is an element d 0 ∈ D such that d 0 ≥ d 1, d 0 ≥ d 2.

  2. Let (D, ≥) be a directed set and X be a non-empty set. A mapping s: D → X is called a net in X, denoted by (s d : d ∈ D) or ( s d ) d ∈ D .

  3. A net ( s d ) d ∈ D is said to be eventually in A ⊆ X if there is d 0 ∈ D such that s d ∈ A for each d ∈ D with d ≥ d 0.

  4. A net ( s d ) d ∈ D in a topological space X is said to converge to x ∈ X if ( s d ) d ∈ D is eventually in every neighborhood of x and we write lim s d = x or (s d ) → x.

Definition 2.7.

[20] Let D be a non-empty set and I be a non-empty subfamily of the power set of D. I is said to be an ideal on D if:

  1. A , B ∈ I implies A ∪ B ∈ I ;

  2. A ∈ I , B ⊆ A implies B ∈ I .

An ideal I is said to be non-trivial if I ≠ { ∅ } and D ∉ I . A non-trivial ideal I is called admissible if it contains all finite subsets of D. Clearly, every non-trivial ideal I defines a dual filter F ( I ) = { A ⊆ D : D \ A ∈ I } on D.

Definition 2.8.

[23],24] Let I be an ideal on a directed set D. A net ( s d ) d ∈ D in a topological space X is said to be I -convergent to a point x ∈ X if for each neighborhood U of x, we have { d ∈ D : s d ∉ U } ∈ I . In this case we write I -lim s d = x or ( s d ) → I x ;

Actually, it is easy to verify that ( s d ) d ∈ D I -converges to x if and only if for each neighborhood U of x, there exists A ∈ F ( I ) such that s d ∈ U for all d ∈ A.

Remark 2.9.

[23],24] Let (D, ≥) be a directed set. For each α ∈ D, define D α = {γ ∈ D: γ ≥ α}.

  1. A non-trivial ideal I of D is said to be D-admissible if D α ∈ F ( I ) for each α ∈ D.

  2. Let I 0 = { A ⊆ D : D \ A ⊇ D α for some α ∈ D } . It is easy to check that I 0 is a D-admissible ideal on D.

  3. If I is a D-admissible ideal on D, ( s d ) d ∈ D is a net in a topological space X and x ∈ X, then (s d ) → x implies ( s d ) → I x and the converse holds if I = I 0 .

  4. If D = N , then the concepts of D-admissibility and admissibility coincide. In this case, I 0 is the ideal of all finite subsets of N .

Analogous to the definition of convergence of sequences in fuzzy metric spaces, a net in a fuzzy metric space (X, M, ∗) is said to be I -convergent if it is I -convergent in (X, τ M ), and a net of functions ( f d ) d ∈ D from a non-empty set X to a fuzzy metric space (Y, M, ∗) is said to be pointwise I -convergent to f ∈ Y X if for each x ∈ X the net ( f d ( x ) ) d ∈ D is I -convergent in (Y, τ M ). In this case, we write ( f d ) d ∈ D → I f or simply ( f d ) → I f and f is called the I -limit function of ( f d ) d ∈ D .

Finally, we recall the definitions of semi-exhaustiveness, semi-α convergence, and semi uniform convergence of sequences of functions defined in [39], which are the main concepts discussed in this paper.

Definition 2.10.

[39] Let (X, d), (Y, ρ) be metric spaces, ( f n ) n ∈ N be a sequence of functions in Y X and f ∈ Y X .

  1. The sequence ( f n ) n ∈ N is said to be semi-exhaustive at x ∈ X if for each ɛ > 0, there exists δ = δ(ɛ, x) > 0 such that for every n ∈ N there exists m ∈ N , m > n such that ρ(f m (y), f m (x)) < ɛ for each y ∈ B(x, δ). The sequence ( f n ) n ∈ N is said to be semi-exhaustive if it is semi-exhaustive at each x ∈ X.

  2. The sequence ( f n ) n ∈ N is said to semi- α converge to f at x if

    1. (f n (x)) → f(x);

    2. for each ɛ > 0 there exists δ = δ(ɛ, x) > 0 such that for every n ∈ N there exists m ∈ N , m ≥ n such that ρ(f m (y), f(x)) < ɛ for each y ∈ B(x, δ).

The sequence ( f n ) n ∈ N is said to be semi- α convergent if it is semi-α convergent at each x ∈ X. A sequence of functions ( f n ) n ∈ N is said to have the semi- α property with respect to f if ( f n ) n ∈ N satisfies the condition (b).

Definition 2.11.

[39] Let X be a topological space, (Y, ρ) be a metric space, ( f n ) n ∈ N be a sequence of functions in Y X and f ∈ Y X . The sequence ( f n ) n ∈ N is said to semi uniformly converge to f at x ∈ X if

  1. (f n (x)) → f(x);

  2. for each ɛ > 0 there exists a neighborhood O of x such that for every n ∈ N there exists m ∈ N , m ≥ n such that ρ(f m (y), f(y)) < ɛ for each y ∈ O.

The sequence ( f n ) n ∈ N is said to be semi uniformly convergent if it is semi uniformly convergent at each x ∈ X.

3 Main results

In this section, D denotes the directed set (D, ≥), I denotes a D-admissible ideal, and (X, N, ⋄) and (Y, M, ∗) denote fuzzy metric spaces, which are simply denoted as X and Y. Nets of functions are always in Y X , unless stated otherwise. A mapping from (X, N, ⋄) to (Y, M, ∗) is called continuous if it is continuous from (X, τ N ) to (Y, τ M ).

3.1 I -semi-α convergence

In this subsection, we define the ideal version of semi-α convergence and clarify their relation. Firstly, the semi-α convergence of nets of functions in fuzzy metric spaces can be naturally defined as follows.

Definition 3.1.

A net of functions ( f d ) d ∈ D is said to semi- α converge to f at x if

  1. (f d (x)) → f(x);

  2. for each ɛ ∈ (0, 1) and t > 0, there exists r ∈ (0, 1) and s > 0 such that for every d ∈ D there exists m ∈ D, m ≥ d such that M(f m (y), f(x), t) > 1 − ɛ for each y ∈ B N (x, r, s).

A net of functions ( f d ) d ∈ D is said to be semi- α convergent if it is semi-α convergent at each x ∈ X. A net of functions ( f d ) d ∈ D is said to have the semi- α property with respect to f if ( f d ) d ∈ D satisfies the condition (2).

Similarly, we can define the semi-exhaustiveness and semi uniform convergence of nets of functions in fuzzy metric spaces.

Let (X, d) and (Y, ρ) be metric spaces. Papanastassiou proved that if a sequence of functions ( f n ) n ∈ N in Y X is pointwise convergent to f and satisfies the semi-α property with respect to f at x ∈ X, then f is continuous at x [39], Proposition 4.3]. It is easy to prove that [39], Proposition 4.3] still holds when X and Y are fuzzy metric spaces and ( f d ) d ∈ D is a net of functions in Y X . We may define I -semi-α convergence by replacing (f d ) → f with ( f d ) → I f in Definition 3.1. However, the following example shows that it fails to preserve the continuity of the pointwise I -limit function in this case.

Example 3.2.

Let D = N and I ≠ I 0 . Then there exists a sequence of functions which is pointwise I -convergent to a function f and satisfies the semi-α property with respect to f, but f is not continuous.

Proof.

Since D = N and I ≠ I 0 , there exists an infinite set A ∈ I . Consider the sequence of functions ( f n ) n ∈ N , where f n : R M | ⋅ | → R M | ⋅ | for each n ∈ N , defined as follows:

f n ( x ) = 1 ,   if n ∈ A ; 1 ,   if n ∉ A , x = 0 ; 1 / n ,   if n ∉ A , x ≠ 0 .

Let f ( x ) = 1 , x = 0 ; 0 , x ≠ 0 .   It is clear that the sequence of functions ( f n ) n ∈ N is pointwise I -convergent to f. Next we show that ( f n ) n ∈ N satisfies the semi-α property with respect to f. For each x ∈ X, if x = 0, fix arbitrarily ɛ > 0, t > 0 and take r = 1 2 , s = 1. For every n ∈ N , since the set A is infinite, there exists m ∈ A with m ≥ n such that M |⋅|(f m (y), f(0), t) = M |⋅|(1, 1, t) = 1 > 1 − ɛ whenever y ∈ B M | ⋅ | ( x , 1 2 , 1 ) . If x ≠ 0, then for each ɛ > 0 and t > 0, there is r ∈ (0, 1) and s > 0 such that 0 ∉ B M | ⋅ | ( x , r , s ) . For every n ∈ N , since the set N \ A is infinite, there is m ∈ N \ A with m ≥ max { [ 1 − ε ε ⋅ t ] + 1 , n } such that M | ⋅ | ( f m ( y ) , f ( x ) , t ) = t t + 1 m > 1 − ε for each y ∈ B M | ⋅ | ( x , r , s ) . It follows that ( f n ) n ∈ N satisfies the semi-α property with respect to f. However, f is not continuous at x = 0.□

For this reason, we define the ideal version of semi-α convergence of nets of functions in fuzzy metric spaces as follows.

Definition 3.3.

A net of functions ( f d ) d ∈ D is said to I -semi- α converge to f at x if

  1. ( f d ( x ) ) → I f ( x ) ;

  2. for each ɛ ∈ (0, 1) and t > 0, there exists r ∈ (0, 1) and s > 0 such that for every A ∈ F ( I ) there exists m ∈ A such that M(f m (y), f(x), t) > 1 − ɛ for each y ∈ B N (x, r, s).

A net of functions ( f d ) d ∈ D is said to be I -semi- α convergent if it is I -semi-α convergent at each x ∈ X. A net of functions ( f d ) d ∈ D is said to have the I -semi- α property with respect to f if it satisfies the condition (2).

For an arbitrary directed set D, I -semi-α convergence coincides with semi-α convergence whenever I = I 0 . However, there exists a directed set D such that for each I ≠ I 0 , they are different. As the following examples show.

Example 3.4.

Let D = N and I ≠ I 0 . Then there exists a sequence of functions which is semi-α convergent but not I -semi-α convergent.

Proof.

Since D = N and I ≠ I 0 , there exists an infinite set A ∈ I . Consider the sequence of functions ( f n ) n ∈ N , where f n : R M | ⋅ | → R M | ⋅ | for each n ∈ N , defined as follows:

f n ( x ) = 1 / n ,   if n ∈ A ; 0 ,   if n ∉ A , x ∈ ( − ∞ , − 1 / n ] ∪ [ 0 , + ∞ ) ; 2 n x + 2 ,   if n ∉ A , x ∈ ( − 1 / n , − 1 / 2 n ] ; − 2 n x ,   if n ∉ A , x ∈ ( − 1 / 2 n , 0 ) .

Let f(x) = 0 for each x ∈ R . For each ɛ ∈ (0, 1) and t > 0, there exists n 1 ∈ N such that M | ⋅ | ( 1 n , 0 , t ) > 1 − ε for every n ≥ n 1. For each x ∈ R , if x ≥ 0, then for every n ≥ n 1 we have M |⋅|(f n (x), f(x), t) > 1 − ɛ; if x < 0, then there exists n 2 ∈ N such that x < − 1 n for every n ≥ n 2. Take n 3 = max{n 1, n 2}. Then for each n ≥ n 3 we have M |⋅|(f n (x), f(x), t) > 1 − ɛ. It follows that ( f n ) n ∈ N is pointwise convergent to f, which implies that it is pointwise I -convergent to f.

In addition, let x ∈ R , ɛ > 0 and t > 0 be given. Choose r ∈ (0, 1) and s > 0. For every n ∈ N , since the set A is infinite, there exists m ∈ A with m ≥ n such that M | ⋅ | ( f m ( y ) , f ( x ) , t ) = M | ⋅ | ( 1 m , 0 , t ) > 1 − ε for each y ∈ B M | ⋅ | ( x , r , s ) . Thus, ( f n ) n ∈ N is semi-α convergent to f.

However, ( f n ) n ∈ N does not I -semi-α converge to f at x 0 = 0. Indeed, take ɛ 0 = 1/2 and t 0 = 1. Then for each r ∈ (0, 1) and s > 0, there exists n 0 ∈ N such that − 1 / 2 n ∈ B M | ⋅ | ( 0 , r , s ) for every n ≥ n 0. For every n ∈ B = ( N \ A ) ∩ { n ∈ N : n ≥ n 0 } ∈ F ( I ) , pick y = − 1 / 2 n ∈ B M | ⋅ | ( 0 , r , s ) . Then we have M |⋅|(f n (y), f(0), t 0) = 1/2 ≤ 1/2. It follows that ( f n ) n ∈ N does not I -semi-α converge to f at x 0 = 0.□

Example 3.5.

Let D = N and I ≠ I 0 . Then there exists an I -semi-α convergent sequence of functions which is not semi-α convergent.

Proof.

Since D = N and I ≠ I 0 , there exists an infinite set A ∈ I . Consider the sequence of functions ( f n ) n ∈ N , where f n : R M | ⋅ | → R M | ⋅ | for each n ∈ N , defined as follows:

f n ( x ) = 1 ,   if n ∈ A ; 1 / n ,   if n ∉ A .

Let f(x) = 0 for each x ∈ R . It is easy to check that ( f n ) n ∈ N is pointwise I -convergent to f. For each x ∈ R , ɛ ∈ (0, 1) and t > 0, there exists n 0 ∈ N such that M | ⋅ | ( 1 n , 0 , t ) > 1 − ε for every n ≥ n 0. Fix arbitrary r ∈ (0, 1) and s > 0. For every B ∈ F ( I ) , let B 1 = B ∩ ( N \ A ) ∩ { n ∈ N : n ≥ n 0 } . Since B 1 ≠ ∅, we can choose m ∈ B 1. Then for each y ∈ B M | ⋅ | ( x , r , s ) , M | ⋅ | ( f m ( y ) , f ( x ) , t ) = M | ⋅ | ( 1 m , 0 , t ) > 1 − ε . It follows that the sequence ( f n ) n ∈ N is I -semi-α convergent to f. However, since the set A is infinite, the sequence of functions ( f n ) n ∈ N is not pointwise convergent to f. Thus it is not semi-α convergent.□

Remark 3.6.

Example 3.4 also shows that a net of functions pointwise I -converging to a continuous function is not necessarily I -semi-α convergent.

3.2 I -semi-exhaustiveness

In this subsection, we consider the ideal version of semi-exhaustiveness.

Definition 3.7.

A net of functions ( f d ) d ∈ D is said to be I -semi-exhaustive at x ∈ X if for each ɛ ∈ (0, 1) and t > 0, there exists r ∈ (0, 1) and s > 0 such that for every A ∈ F ( I ) there exists m ∈ A such that M(f m (y), f m (x), t) > 1 − ɛ for each y ∈ B N (x, r, s). The net ( f d ) d ∈ D is said to be I -semi-exhaustive if it is I -semi-exhaustive at each x ∈ X.

For an arbitrary directed set D, it is easy to check that I -semi-exhaustiveness implies semi-exhaustiveness, and I -semi-exhaustiveness coincides with semi-exhaustiveness whenever I = I 0 . However, there exists a directed set D such that for each I ≠ I 0 , there is a net of functions which is semi-exhaustive but not I -semi-exhaustive.

Example 3.8.

Let D = N and I ≠ I 0 . Then there is a semi-exhaustive sequence of functions which is not I -semi-exhaustive.

Proof.

Since D = N and I ≠ I 0 , there exists an infinite set A ∈ I . Consider the sequence of functions ( f n ) n ∈ N , where f n : R M | ⋅ | → R M | ⋅ | for each n ∈ N , defined as follows:

f n ( x ) = 1 ,   if n ∈ A ; 1 ,   if n ∉ A , x < 0 ; 1 / n ,   if n ∉ A , x ≥ 0 .

Let x ∈ R , ɛ > 0 and t > 0 be given. Choose r ∈ (0, 1) and s > 0. For every n ∈ N , since the set A is infinite, there is m ∈ A with m ≥ n such that M |⋅|(f m (y), f m (x), t) = 1 > 1 − ɛ for each y ∈ B M | ⋅ | ( x , r , s ) . It follows that ( f n ) n ∈ N is semi-exhaustive. However, take ɛ = 1/2 and t 0 = 1/2. For each r ∈ (0, 1), s > 0 and n ∈ N \ ( A ∪ { 1 } ) , pick y 0 ∈ B M | ⋅ | ( 0 , r , s ) ∩ ( − ∞ , 0 ) . Then, M | ⋅ | ( f n ( y 0 ) , f n ( 0 ) , 1 2 ) = n 3 n − 2 ≤ 1 / 2 . Since the set N \ ( A ∪ { 1 } ) ∈ F ( I ) , it follows that ( f n ) n ∈ N is not I -semi-exhaustive at x 0 = 0.□

The following example shows that there is a net of functions which is pointwise I -convergent to a continuous function but not I -semi-exhaustive.

Example 3.9.

For an arbitrary ideal I , there is a net of functions which is pointwise I -convergent to a continuous function but not I -semi-exhaustive.

Proof.

Let D = N , consider the sequence of functions ( f n ) n ∈ N , where f n : R M | ⋅ | → R M | ⋅ | for each n ∈ N , defined as follows:

f n ( x ) = 1 / 4 n ,   if x ≤ 0 ; n ,   if x = 1 / n ; 1 / n ,   if x > 0 , x ≠ 1 / n .

It is easy to check that ( f n ) n ∈ N is pointwise I -convergent to f ≡ 0. However, take ɛ = 1/4 and t 0 = 1. For each r ∈ (0, 1) and s > 0, there is n 0 ∈ N such that 1 n ∈ B M | ⋅ | ( 0 , r , s ) for each n ≥ n 0. Let A = { n ∈ N : n ≥ n 0 } . Then for each n ∈ A, there is y = 1 n ∈ B M | ⋅ | ( 0 , r , s ) such that

M | ⋅ | ( f n ( y ) , f n ( 0 ) , t 0 ) = 1 1 + n − 1 4 n ≤ 3 4 .

It follows that ( f n ) n ∈ N is not I -semi-exhaustive at x 0 = 0.□

Let D be the class of all D-admissible ideals on (D, ≥). The class D is partially ordered by inclusion. If D 0 ⊆ D is a non-empty linearly ordered subset of D , then it is easy to verify that ∪ D 0 is a D-admissible ideal on (D, ≥) which is an upper bound of D 0 . It follows from the Zorn’s Lemma that there is a maximal D-admissible ideal on (D, ≥). The following lemma gives a characterization of a maximal D-admissible ideal.

Lemma 3.10.

Let I be a D-admissible ideal on (D, ≥). Then I is a maximal D-admissible ideal if and only if

( A ∈ I ) ∨ ( D \ A ∈ I )

holds for each A ⊆ D.

Proof.

(⇐) Let a D-admissible ideal I fulfill the condition for each A ⊆ D. We show that I is a maximal D-admissible ideal. On the contrary, let I be a proper subset of I 1 where I 1 is a D-admissible ideal on D. Then, there is A ⊆ D such that A ∈ ( I 1 \ I ) . Since A ∉ I , by assumption we have ( D \ A ) ∈ I . Consequently A ∈ I 1 and ( D \ A ) ∈ I 1 , so D ∈ I 1 , which is a contradiction.

(⇒) Let I be a maximal D-admissible ideal. If possible, let A ⊆ D be such that

( A ∉ I ) ∧ ( D \ A ∉ I ) .

Define K = { E ⊆ D : E ∩ A ∈ I } . For each B ∈ I , since B ∩ A ⊆ B ∈ I , it follows that B ∩ A ∈ I , and hence B ∈ K . Thus I ⊆ K . Obviously D ∉ K . It follows immediately that K is a non-trivial D-admissible ideal containing I . However, I is maximal, so I = K . Now since ( D \ A ) ∩ A = ∅ ∈ I , this shows that ( D \ A ) ∈ I which is again a contradiction. This proves the lemma.□

The ideal version of exhaustiveness of sequences of functions is defined in [37] between metric spaces, and it can be extended to nets of functions between fuzzy metric spaces naturally as follows.

Definition 3.11.

A net of functions ( f d ) d ∈ D is said to be I -exhaustive at x ∈ X if for each ɛ ∈ (0, 1) and t > 0, there exists r ∈ (0, 1), s > 0 and A ∈ F ( I ) such that for every m ∈ A and y ∈ B N (x, r, s) we have M(f m (y), f m (x), t) > 1 − ɛ. The net ( f d ) d ∈ D is said to be I -exhaustive if it is I -exhaustive at each x ∈ X.

Next we show that I -exhaustiveness and I -semi-exhaustiveness are equivalent when I is a maximal D-admissible ideal.

Theorem 3.12.

If I is a maximal D-admissible ideal, then a net of functions ( f d ) d ∈ D is I -semi-exhaustive if and only if it is I -exhaustive.

Proof.

(⇐) It is clear by the definition of I -semi-exhaustiveness and I -exhaustiveness.

(⇒) Since ( f d ) d ∈ D is I -semi-exhaustive, for each x ∈ X, ɛ ∈ (0, 1) and t > 0, there exists r ∈ (0, 1) and s > 0 such that for every A ∈ F ( I ) there exists m ∈ A such that M(f m (y), f m (x), t) > 1 − ɛ for each y ∈ B N (x, r, s). Let

H = { d ∈ D : M ( f d ( y ) , f d ( x ) , t ) > 1 − ε for each y ∈ B N ( x , r , s ) } .

Then M(f d (y), f d (x), t) > 1 − ɛ for each d ∈ H and y ∈ B N (x, r, s). We claim that H ∈ F ( I ) . Suppose that H ∉ F ( I ) , since I is maximal, this means that D \ H ∈ F ( I ) . Therefore, there exists m ∈ D \ H such that M(f m (y), f m (x), t) > 1 − ɛ for all y ∈ B N (x, r, s). By the construction of H, we can conclude that m ∈ H, which is a contradiction. Hence, ( f d ) d ∈ D is I -exhaustive at x, and then ( f d ) d ∈ D is I -exhaustive.□

3.3 I -semi uniform convergence

In this subsection, we define the ideal version of semi uniform convergence of nets of functions, and prove the main results of this paper.

Definition 3.13.

A net of functions ( f d ) d ∈ D is said to I -semi uniformly converge to f at x if

  1. ( f d ( x ) ) → I f ( x ) ;

  2. for each ɛ ∈ (0, 1) and t > 0, there exists r ∈ (0, 1) and s > 0 such that for every A ∈ F ( I ) there exists m ∈ A such that M(f m (y), f(y), t) > 1 − ɛ for each y ∈ B N (x, r, s).

The net ( f d ) d ∈ D is said to be I -semi uniformly convergent if it is I -semi uniformly convergent at each x ∈ X.

For an arbitrary directed set D, from the definitions of I -semi uniform convergence and semi uniform convergence, it is easy to check that when I = I 0 , I -semi uniform convergence coincides with semi uniform convergence.

Next we will show that I -semi uniform convergence preserves the continuity of the pointwise I -limit of a net of continuous functions.

Theorem 3.14.

Let ( f d ) d ∈ D be a net of functions in C(X, Y) and f ∈ Y X . If ( f d ) d ∈ D is I -semi uniformly convergent to f, then f is continuous.

Proof.

For each ɛ ∈ (0, 1), there is ɛ 1 ∈ (0, ɛ) such that (1 − ɛ 1)∗(1 − ɛ 1)∗(1 − ɛ 1) > 1 − ɛ. For each x ∈ X and t > 0, since ( f d ) d ∈ D is I -semi uniformly convergent to f at x, there exists r 1 ∈ (0, 1) and s 1 > 0 such that for every A ∈ F ( I ) there exists m ∈ A such that for each y ∈ B N (x, r 1, s 1) we have

M f m ( y ) , f ( y ) , t 3 > 1 − ε 1 .

As the net of functions ( f d ) d ∈ D pointwise I -converges to f at x, there is A ′ ∈ F ( I ) such that M ( f d ( x ) , f ( x ) , t 3 ) > 1 − ε 1 for every d ∈ A′. Thus there is m ∈ A′ such that

M f m ( y ) , f ( y ) , t 3 > 1 − ε 1 , M f m ( x ) , f ( x ) , t 3 > 1 − ε 1 .

By the continuity of f m , there exists r 2 ∈ (0, 1) and s 2 > 0 such that

M f m ( y ) , f m ( x ) , t 3 > 1 − ε 1

for each y ∈ B N (x, r 2, s 2). Let r = min{r 1, r 2} and s = min{s 1, s 2}. Then for each y ∈ B N (x, r, s) we have

M ( f ( y ) , f ( x ) , t ) ≥ M f m ( y ) , f ( y ) , t 3 ∗ M f m ( x ) , f ( x ) , t 3 ∗ M f m ( y ) , f m ( x ) , t 3

≥ ( 1 − ε 1 ) ∗ ( 1 − ε 1 ) ∗ ( 1 − ε 1 ) > 1 − ε .

Hence f is continuous at x, and then f is continuous.□

The condition ( f d ) d ∈ D in C(X, Y) is necessary in Theorem 3.14 as the following example states.

Example 3.15.

For an arbitrary ideal I , there is a net of functions ( f d ) d ∈ D I -semi uniformly converging to a function f, which is not continuous.

Proof.

Let D = N . Consider the sequence of functions ( f n ) n ∈ N , where f n : R M | ⋅ | → R M | ⋅ | for each n ∈ N , defined as follows:

f n ( x ) = 1 ,   if x = 0 ; 1 / n ,   if x ≠ 0 .

Let f ( x ) = 1 , x = 0 ; 0 , x ≠ 0 .   It is clear that ( f n ) n ∈ N is pointwise convergent to f, which implies that it is pointwise I -convergent to f. For each x ∈ X, ɛ, r ∈ (0, 1), t, s > 0 and A ∈ F ( I ) , since A is infinite, there exists m ∈ A such that M | ⋅ | ( 1 m , 0 , t ) > 1 − ε . Then for each y ∈ B M | ⋅ | ( x , r , s ) , if y = 0, we have M |⋅|(f m (y), f(y), t) = 1 > 1 − ɛ; if y ≠ 0, we have M | ⋅ | ( f m ( y ) , f ( y ) , t ) = M | ⋅ | ( 1 m , 0 , t ) > 1 − ε . Thus, ( f n ) n ∈ N is I -semi uniformly convergent to f. However, f is not continuous.□

The following result reveals the relations among I -semi-exhaustiveness, I -semi-α convergence, and I -semi uniform convergence of a net of functions.

Theorem 3.16.

Let ( f d ) d ∈ D be a net of functions in Y X and f ∈ Y X . Then the following are equivalent:

  1. The net ( f d ) d ∈ D is I -semi-exhaustive and ( f d ) → I f .

  2. The net ( f d ) d ∈ D I -semi-α converges to f.

  3. The net ( f d ) d ∈ D I -semi uniformly converges to f and the function f is continuous.

Proof.

(1) ⇒ (2) For each ɛ ∈ (0, 1), there is ɛ 1 ∈ (0, ɛ) such that (1 − ɛ 1)∗(1 − ɛ 1) > 1 − ɛ. For each x ∈ X and t > 0, since ( f d ) d ∈ D is I -semi-exhaustive at x, there exists r ∈ (0, 1) and s > 0 such that for every A ∈ F ( I ) there is m ∈ A such that

M f m ( y ) , f m ( x ) , t 2 > 1 − ε 1

for each y ∈ B N (x, r, s). By assumption, ( f d ) d ∈ D is pointwise I -convergent to f, then there exists A ′ ∈ F ( I ) such that

M f d ( x ) , f ( x ) , t 2 > 1 − ε 1

for every d ∈ A′. Hence, for every A ∈ F ( I ) , since A ∩ A ′ ∈ F ( I ) , there is m ∈ A ∩ A′ such that

M f m ( y ) , f m ( x ) , t 2 > 1 − ε 1 , M f m ( x ) , f ( x ) , t 2 > 1 − ε 1

for each y ∈ B N (x, r, s). It follows that

M ( f m ( y ) , f ( x ) , t ) ≥ M f m ( y ) , f m ( x ) , t 2 ∗ M f m ( x ) , f ( x ) , t 2

≥ ( 1 − ε 1 ) ∗ ( 1 − ε 1 ) > 1 − ε

for each y ∈ B N (x, r, s). Therefore, the net ( f d ) d ∈ D is I -semi-α convergent to f at x, and then it is I -semi-α convergent to f.

(2) ⇒ (3) First we prove that f is continuous. For each ɛ ∈ (0, 1), there is ɛ 1 ∈ (0, ɛ) such that (1 − ɛ 1)∗(1 − ɛ 1) > 1 − ɛ. For each x ∈ X and t > 0, since the net ( f d ) d ∈ D satisfies the I -semi-α property at x with respect to f, there exists r 1 ∈ (0, 1) and s 1 > 0 such that for every A ∈ F ( I ) there is m ∈ A such that

M f m ( y ) , f ( x ) , t 2 > 1 − ε 1

for each y ∈ B N (x, r 1, s 1). Since ( f d ) d ∈ D pointwise I -converges to f, there exists A y ∈ F ( I ) such that

M f d ( y ) , f ( y ) , t 2 > 1 − ε 1

for every d ∈ A y . Let A ∈ F ( I ) . Then, for each y ∈ B N (x, r 1, s 1), there is m 1 ∈ A ∩ A y such that

M ( f ( y ) , f ( x ) , t ) ≥ M f m 1 ( y ) , f ( x ) , t 2 ∗ M f m 1 ( y ) , f ( y ) , t 2

≥ ( 1 − ε 1 ) ∗ ( 1 − ε 1 ) > 1 − ε .

It follows that f is continuous at x, then f is continuous.

It remains to prove that the net ( f d ) d ∈ D is I -semi uniformly convergent to f. Clearly, (2) implies that ( f d ) → I f . For each x ∈ X and t > 0, since the net ( f d ) d ∈ D satisfies the I -semi-α property at x with respect to f, there exists r 1 ∈ (0, 1) and s 1 > 0 such that for every A ∈ F ( I ) there is m ∈ A such that

M f m ( y ) , f ( x ) , t 2 > 1 − ε 1

for all y ∈ B N (x, r 1, s 1). By the continuity of f, there exists r 2 ∈ (0, 1) and s 2 > 0 such that

M f ( y ) , f ( x ) , t 2 > 1 − ε 1

for each y ∈ B N (x, r 2, s 2). Let r = min{r 1, r 2} and s = min{s 1, s 2}. Then for each y ∈ B N (x, r, s) we have

M ( f m ( y ) , f ( y ) , t ) ≥ M f m ( y ) , f ( x ) , t 2 ∗ M f ( y ) , f ( x ) , t 2

≥ ( 1 − ε 1 ) ∗ ( 1 − ε 1 ) > 1 − ε .

It follows that the net is I -semi uniformly convergent to f at x, then it is I -semi uniformly convergent to f.

(3) ⇒ (1) It is clear that (3) implies that ( f d ) → I f . For each ɛ ∈ (0, 1), there is ɛ 1 ∈ (0, ɛ) such that (1 − ɛ 1)∗(1 − ɛ 1)∗(1 − ɛ 1) > 1 − ɛ. For each x ∈ X and t > 0, since ( f d ) d ∈ D is I -semi uniformly convergent to f at x, there exists r 1 ∈ (0, 1) and s 1 > 0 such that for every A ∈ F ( I ) there exists m ∈ A such that

M f m ( y ) , f ( y ) , t 3 > 1 − ε 1

for all y ∈ B N (x, r 1, s 1). By the continuity of f, there exists r 2 ∈ (0, 1) and s 2 > 0 such that

M f ( y ) , f ( x ) , t 3 > 1 − ε 1

for all y ∈ B N (x, r 2, s 2). Let r = min{r 1, r 2} and s = min{s 1, s 2}. Then for each y ∈ B N (x, r, s) we have

M ( f m ( y ) , f m ( x ) , t ) ≥ M f m ( y ) , f ( y ) , t 3 ∗ M f ( y ) , f ( x ) , t 3 ∗ M f m ( x ) , f ( x ) , t 3

≥ ( 1 − ε 1 ) ∗ ( 1 − ε 1 ) ∗ ( 1 − ε 1 ) > 1 − ε .

Thus the net is I -semi-exhaustive at x, then it is I -semi-exhaustive.□

Remark 3.17.

By Theorem 3.16, Example 3.4 also shows that there exists a directed set D such that I -semi uniform convergence and semi uniform convergence are different whenever I ≠ I 0 .

Let D = N and I ≠ I 0 . The sequences of functions ( f n ) n ∈ N defined in Example 3.8 is semi-exhaustive and pointwise I -convergent to f ( x ) = 1 , x < 0 ; 0 , x ≥ 0 .   Clearly, f is not continuous at x 0 = 0. This shows that semi-exhaustiveness is too weak to preserve the continuity of the pointwise I -limit function. However, by Theorems 3.16, we have the following corollaries.

Corollary 3.18.

If a net of functions ( f d ) d ∈ D is I -semi-exhaustive and ( f d ) → I f ∈ Y X , then f is continuous.

Corollary 3.19.

If a net of functions ( f d ) d ∈ D I -semi-α converges to f ∈ Y X , then f is continuous.

4 Conclusions

Recently, N. Papanastassiou [39] introduced the concepts of semi-α convergence, semi-exhaustiveness and semi uniform convergence of sequences of functions between metric spaces. Since fuzzy metric spaces can be regarded as a generalization of metric spaces in a certain sense, it is natural to consider whether these new concepts can be extended to fuzzy metric spaces and which properties of them remain invariant. Note that I -convergence of nets implies convergence of sequences, when the directed set is N and I = I 0 . This motivates us to define appropriate ideal versions of semi-α convergence, semi-exhaustiveness and semi uniform convergence for nets of functions such that they imply the classical one when the directed set is N and I = I 0 , and to study them in the modified way.

Thus, in this paper, we define and study ideal versions of semi-α convergence, semi-exhaustiveness and semi uniform convergence of nets of functions in fuzzy metric spaces. Our work extends some corresponding results in [39]. We also prove that when I is a maximal D-admissible ideal, then a net of functions is I -exhaustive if and only if it is I -semi-exhaustive. However, ideal versions of α-convergence and almost uniform convergence of nets of functions in fuzzy metric spaces have not been considered in this paper. Thus, one can define ideal versions of α-convergence and almost uniform convergence of nets of functions in fuzzy metric spaces, and study their properties or their relations with the corresponding concepts.

Concluding this section, we briefly mention another related study on I -convergence. In 2024, the first and third authors studied the properties of I -convergence in cone metric spaces [42]. The paper discusses I -sequential compactness, I -sequential countable compactness, and I -completeness in cone metric spaces, and provides a negative answer to an open problem posed by P. Das [24], Open Problem 2.3]. Both reference [42] and this paper concern I -convergence, but they focus on different specific aspects. As a direction for future research, we can study I -sequential compactness, sequential countable compactness, and completeness in fuzzy metric spaces.

Corresponding author: Zhongbao Tang, School of Mathematics and Statistics, Minnan Normal University, 363000 Zhangzhou, P. R. China, E-mail: 

Acknowledgments

The authors sincerely thank the reviewers for their careful reading and valuable suggestions in improving this manuscript.

  1. Research ethics: Not applicable.

  2. Informed consent: Not applicable.

  3. Author contributions: All authors have accepted responsibility for the entire content of this manuscript and consented to its submission to the journal, reviewed all the results and approved the final version of the manuscript. All authors contributed equally to this work.

  4. Use of Large Language Models, AI and Machine Learning Tools: None declared.

  5. Conflict of interest: The authors state no conflicts of interest.

  6. Research funding: This research was supported by National Natural Science Foundation of China (No.∼11901274), the Fujian Provincial Natural Science Foundation of China (Nos.∼2025J01360,∼2024J02022, ∼2024J01804), the Natural Science Foundation of Jiangsu Province (No. ∼BK20200834), the Institute of Fujian Key Laboratory of Granular Computing and Applications, the Institute of Meteorological Big Data-Digital Fujian, Fujian Key Laboratory of Data Science and Statistics, and President’s fund of Minnan Normal University (KJ18007).

  7. Data availability: Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

References

[1] L. A. Zadeh, Fuzzy sets, Inf. Control 8 (1965), no. 3, 338–353. https://doi.org/10.1016/s0019-9958(65)90241-x.Search in Google Scholar

[2] A. George and P. Veeramani, On some results in fuzzy metric spaces, Fuzzy Sets Syst. 64 (1994), no. 3, 395–399. https://doi.org/10.1016/0165-0114(94)90162-7.Search in Google Scholar

[3] I. Kramosil and J. Michálek, Fuzzy metric and statistical metric spaces, Kybernetika 11 (1975), no. 5, 336–344.Search in Google Scholar

[4] V. Gregori and S. Romaguera, Some properties of fuzzy metric spaces, Fuzzy Sets Syst. 115 (2000), no. 3, 485–489. https://doi.org/10.1016/s0165-0114(98)00281-4.Search in Google Scholar

[5] A. George and P. Veeramani, On some results of analysis for fuzzy metric spaces, Fuzzy Sets Syst. 90 (1997), no. 3, 365–368. https://doi.org/10.1016/s0165-0114(96)00207-2.Search in Google Scholar

[6] C. Li and K. Li, On topological properties of the Hausdorff fuzzy metric spaces, Filomat 31 (2017), no. 5, 1167–1173. https://doi.org/10.2298/fil1705167l.Search in Google Scholar

[7] C. Li and Y. Zhang, On p-convergent sequences and p-Cauchy sequences in fuzzy metric spaces, Fuzzy Sets Syst. 466 (2023), 108464. https://doi.org/10.1016/j.fss.2022.12.020.Search in Google Scholar

[8] A. Zygmund, Trigonometric Series, Vol. I, II, 3rd ed., With a foreword by Robert A. Fefferman, Cambridge University Press, Cambridge, 2002.10.1017/CBO9781316036587Search in Google Scholar

[9] H. Fast, Sur la convergence statistique, Colloq. Math. 2 (1951), 241–244. https://doi.org/10.4064/cm-2-3-4-241-244.Search in Google Scholar

[10] H. Steinhaus, Sur la convergence ordinaire et la convergence asymptotique, Colloq. Math. 2 (1951), no. 1, 73–74.Search in Google Scholar

[11] T. Šalát, On statistically convergent sequences of real numbers, Math. Slovaca 30 (1980), no. 2, 139–150.Search in Google Scholar

[12] J. A. Fridy, On statistical convergence, Analysis 5 (1985), no. 4, 301–314. https://doi.org/10.1524/anly.1985.5.4.301.Search in Google Scholar

[13] J. A. Fridy, Statistical limit points, Proc. Am. Math. Soc. 118 (1993), 1187–1192. https://doi.org/10.2307/2160076.Search in Google Scholar

[14] G. Di Maio and L. D. R. Kočinac, Statistical convergence in topology, Topol. Appl. 156 (2008), no. 1, 28–45. https://doi.org/10.1016/j.topol.2008.01.015.Search in Google Scholar

[15] H. Çakallı, On statistical convergence in topological groups, Pure Appl. Math. Sci. 43 (1996), no. 1–2, 27–31.Search in Google Scholar

[16] A. Caserta and L. D. R. Kočinac, On statistical exhaustiveness, Appl. Math. Lett. 25 (2012), no. 10, 1447–1451. https://doi.org/10.1016/j.aml.2011.12.022.Search in Google Scholar

[17] C. Li, Y. Zhang, and J. Zhang, On statistical convergence in fuzzy metric spaces, J. Intell. Fuzzy Syst. 39 (2020), no. 3, 3987–3993. https://doi.org/10.3233/jifs-200148.Search in Google Scholar

[18] K. Li, S. Lin, and Y. Ge, On statistical convergence in cone metric spaces, Topol. Appl. 196 (2015), 641–651. https://doi.org/10.1016/j.topol.2015.05.038.Search in Google Scholar

[19] Z. Tang and F. Lin, Statistical versions of sequential and Fréchet-Urysohn spaces, Adv. Math. 44 (2015), no. 6, 945–954.Search in Google Scholar

[20] P. Kostyrko, T. Šalát, and W. Wilczyński, I-convergence, Real Anal. Exchange 26 (2001), no. 2, 669–686.10.2307/44154069Search in Google Scholar

[21] M. Balcerzak, M. Poplawski, and A. Wachowicz, The Baire category of ideal convergent subseries and rearrangements, Topol. Appl. 231 (2017), 219–230.10.1016/j.topol.2017.09.014Search in Google Scholar

[22] P. Das, Certain types of open covers and selection principles using ideals, Houston J. Math. 39 (2013), no. 2, 637–650.Search in Google Scholar

[23] B. K. Lahiri and P. Das, I and I*-convergence of nets, Real Anal. Exchange 33 (2008), no. 2, 431–442.10.14321/realanalexch.33.2.0431Search in Google Scholar

[24] P. Das, Summability and convergence using ideals, in: H. Dutta and B. E. Rhoades (eds), Current Topics in Summability Theory and Applications, Springer-Verlag, Singapore, 2016, pp. 77–140.10.1007/978-981-10-0913-6_3Search in Google Scholar

[25] S. Lin, On I-neighborhood spaces and I-quotient spaces, Bull. Malays. Math. Sci. Soc. 44 (2021), no. 4, 1979–2004. https://doi.org/10.1007/s40840-020-01043-1.Search in Google Scholar

[26] Z. Tang and Q. Xiong, A note on I-convergence in quasi-metric spaces, Filomat 37 (2023), no. 4, 1133–1142. https://doi.org/10.2298/fil2304133t.Search in Google Scholar

[27] X. Zhou and S. Lin, On I-covering images of metric spaces, Filomat 36 (2022), no. 19, 6621–6629. https://doi.org/10.2298/fil2219621z.Search in Google Scholar

[28] X. Zhou, S. Lin, and H. Zhang, Isn-sequential spaces and the images of metric spaces, Topol. Appl. 327 (2023), 108439. https://doi.org/10.1016/j.topol.2023.108439.Search in Google Scholar

[29] U. Dini and T. Tipografia, Fondamenti per la teoria delle funzioni di variabili reali, Nistri e Compagni, Pisa, 1878.Search in Google Scholar

[30] C. Arzelà, Intorno alla continuità della somma di infinite di funzioni continue, Rend. R. Accad. Sci. Istit. Bologna (1883–1884), 79–84.Search in Google Scholar

[31] É. Borel, Leçons sur les functions de variables réelles et les développements en séries de polynômes, Gauthier-Villars, Paris, 1905.Search in Google Scholar

[32] R. F. Arens, A topology for spaces of transformations, Ann. Math. 47 (1946), no. 3, 480–495. https://doi.org/10.2307/1969087.Search in Google Scholar

[33] H. Hahn, Reelle Funktionen, Chelsea Publishing Company, New York, 1948.Search in Google Scholar

[34] S. Stoilov, Continuous convergence, Rev. Math. Pures Appl. 4 (1959), 341–344.Search in Google Scholar

[35] K. Iséki, A theorem on continuous convergence, Proc. Japan Acad. 33 (1957), no. 7, 355–356. https://doi.org/10.3792/pja/1195524988.Search in Google Scholar

[36] V. Gregoriades and N. Papanastassiou, The notion of exhaustiveness and Ascoli-type theorems, Topol. Appl. 155 (2008), no. 10, 1111–1128.10.1016/j.topol.2008.02.005Search in Google Scholar

[37] Ch. Papachristodoulos, N. Papanastassiou, and W. Wilczyński, I-exhaustive sequences of functions, in: Selected Papers of the ICTA, 2010.Search in Google Scholar

[38] A. C. Megaritis, Ideal convergence of nets of functions with values in uniform spaces, Filomat 31 (2017), no. 20, 6281–6292. https://doi.org/10.2298/fil1720281m.Search in Google Scholar

[39] N. Papanastassiou, A note on convergence of sequences of functions, Topol. Appl. 275 (2020), 107017. https://doi.org/10.1016/j.topol.2019.107017.Search in Google Scholar

[40] R. Engelking, General Topology (revised and completed edition), Heldermann Verlag, Berlin, 1989.Search in Google Scholar

[41] M. Grabiec, Fixed points in fuzzy metric spaces, Fuzzy Sets Syst. 27 (1988), no. 3, 385–389. https://doi.org/10.1016/0165-0114(88)90064-4.Search in Google Scholar

[42] L. Zhong and Z. Tang, Some properties of I-convergence in cone metric spaces, Filomat 38 (2024), no. 16, 5819–5826.https://doi.org/10.2298/fil2416819z.Search in Google Scholar
Received: 2025-02-24
Accepted: 2025-10-30
Published Online: 2025-12-22

© 2025 the author(s), published by De Gruyter, Berlin/Boston

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

  1. On I-convergence of nets of functions in fuzzy metric spaces
  2. Special Issue on Contemporary Developments in Graphs Defined on Algebraic Structures
  3. Forbidden subgraphs of TI-power graphs of finite groups
  4. Finite group with some c#-normal and S-quasinormally embedded subgroups
  5. Classifying cubic symmetric graphs of order 88p and 88p 2
  6. Two-sided zero-divisor graphs of orientation-preserving and order-decreasing transformation semigroups
  7. Simplicial complexes defined on groups
  8. Further results on permanents of Laplacian matrices of trees
  9. Algebra
  10. Classes of modules closed under projective covers
  11. On the dimension of the algebraic sum of subspaces
  12. Green's graphs of a semigroup
  13. On an uncertainty principle for small index subgroups of finite fields
  14. On a generalization of I-regularity
  15. Algorithm and linear convergence of the H-spectral radius of weakly irreducible quasi-positive tensors
  16. The hyperbolic CS decomposition of tensors based on the C-product
  17. On weakly classical 1-absorbing prime submodules
  18. Equational characterizations for some subclasses of domains
  19. Algebraic Geometry
  20. Spin(8, ℂ)-Higgs bundles fixed points through spectral data
  21. Embedding of lattices and K3-covers of an Enriques surface
  22. Kodaira-Spencer maps for elliptic orbispheres as isomorphisms of Frobenius algebras
  23. Applications in Computer and Information Sciences
  24. Dynamics of particulate emissions in the presence of autonomous vehicles
  25. Exploring homotopy with hyperspherical tracking to find complex roots with application to electrical circuits
  26. Category Theory
  27. The higher mapping cone axiom
  28. Combinatorics and Graph Theory
  29. 𝕮-inverse of graphs and mixed graphs
  30. On the spectral radius and energy of the degree distance matrix of a connected graph
  31. Some new bounds on resolvent energy of a graph
  32. Coloring the vertices of a graph with mutual-visibility property
  33. Ring graph induced by a ring endomorphism
  34. A note on the edge general position number of cactus graphs
  35. Complex Analysis
  36. Some results on value distribution concerning Hayman's alternative
  37. Freely quasiconformal and locally weakly quasisymmetric mappings in metric spaces
  38. A new result for entire functions and their shifts with two shared values
  39. On a subclass of multivalent functions defined by generalized multiplier transformation
  40. Singular direction of meromorphic functions with finite logarithmic order
  41. Growth theorems and coefficient bounds for g-starlike mappings of complex order λ
  42. Refinements of inequalities on extremal problems of polynomials
  43. Control Theory and Optimization
  44. Averaging method in optimal control problems for integro-differential equations
  45. On superstability of derivations in Banach algebras
  46. The robust isolated calmness of spectral norm regularized convex matrix optimization problems
  47. Observability on the classes of non-nilpotent solvable three-dimensional Lie groups
  48. Differential Equations
  49. The ill-posedness of the (non-)periodic traveling wave solution for the deformed continuous Heisenberg spin equation
  50. A note on the global existence and boundedness of an N-dimensional parabolic-elliptic predator-prey system with indirect pursuit-evasion interaction
  51. Blow-up of solutions for Euler-Bernoulli equation with nonlinear time delay
  52. Periodic or homoclinic orbit bifurcated from a heteroclinic loop for high-dimensional systems
  53. Regularity of weak solutions to the 3D stationary tropical climate model
  54. Local minimizers for the NLS equation with localized nonlinearity on noncompact metric graphs
  55. Global existence and blow-up of solutions to pseudo-parabolic equation for Baouendi-Grushin operator
  56. Bubbles clustered inside for almost-critical problems
  57. Existence and multiplicity of positive solutions for multiparameter periodic systems
  58. Existence of positive periodic solutions for evolution equations with delay in ordered Banach spaces
  59. On a nonlinear boundary value problems with impulse action
  60. Normalized ground-states for the Sobolev critical Kirchhoff equation with at least mass critical growth
  61. Multiple positive solutions to a p-Kirchhoff equation with logarithmic terms and concave terms
  62. Infinitely many solutions for a class of Kirchhoff-type equations
  63. Real and non-real eigenvalues of regular indefinite Sturm–Liouville problems
  64. Existence of global solutions to a semilinear thermoelastic system in three dimensions
  65. Limiting profile of positive solutions to heterogeneous elliptic BVPs with nonlinear flux decaying to negative infinity on a portion of the boundary
  66. Morse index of circular solutions for repulsive central force problems on surfaces
  67. Differential Geometry
  68. On tangent bundles of Walker four-manifolds
  69. Pedal and negative pedal surfaces of framed curves in the Euclidean 3-space
  70. Discrete Mathematics
  71. Eventually monotonic solutions of the generalized Fibonacci equations
  72. Dynamical Systems Ergodic Theory
  73. Dynamical properties of two-diffusion SIR epidemic model with Markovian switching
  74. A note on weighted measure-theoretic pressure
  75. Pullback attractors for a class of second-order delay evolution equations with dispersive and dissipative terms on unbounded domain
  76. Pullback attractor of the 2D non-autonomous magneto-micropolar fluid equations
  77. Functional Analysis
  78. Spectrum boundary domination of semiregularities in Banach algebras
  79. Approximate multi-Cauchy mappings on certain groupoids
  80. Investigating the modified UO-iteration process in Banach spaces by a digraph
  81. Tilings, sub-tilings, and spectral sets on p-adic space
  82. Continuity and essential norm of operators defined by infinite tridiagonal matrices in weighted Orlicz and l∞ spaces
  83. A family of commuting contraction semigroups on l 1 ( N ) and l ∞ ( N )
  84. q-Stirling sequence spaces associated with q-Bell numbers
  85. Chlodowsky variant of Bernstein-type operators on the domain
  86. Hyponormality on a weighted Bergman space of an annulus with a general harmonic symbol
  87. Characterization of derivations on strongly double triangle subspace lattice algebras by local actions
  88. Fixed point approaches to the stability of Jensen’s functional equation
  89. Geometry
  90. The regularity of solutions to the Lp Gauss image problem
  91. Solving the quartic by conics
  92. Group Theory
  93. On a question of permutation groups acting on the power set
  94. A characterization of the translational hull of a weakly type B semigroup with E-properties
  95. Harmonic Analysis
  96. Eigenfunctions on an infinite Schrödinger network
  97. Maximal function and generalized fractional integral operators on the weighted Orlicz-Lorentz-Morrey spaces
  98. Subharmonic functions and associated measures in ℝn
  99. Mathematical Logic, Model Theory and Foundation
  100. A topology related to implication and upsets on a bounded BCK-algebra
  101. Boundedness of fractional sublinear operators on weighted grand Herz-Morrey spaces with variable exponents
  102. Number Theory
  103. Fibonacci vector and matrix p-norms
  104. Recurrence for probabilistic extension of Dowling polynomials
  105. Carmichael numbers composed of Piatetski-Shapiro primes in Beatty sequences
  106. The number of rational points of some classes of algebraic varieties over finite fields
  107. Classification and irreducibility of a class of integer polynomials
  108. Decompositions of the extended Selberg class functions
  109. Joint approximation of analytic functions by the shifts of Hurwitz zeta-functions in short intervals
  110. Fibonacci Cartan and Lucas Cartan numbers
  111. Recurrence relations satisfied by some arithmetic groups
  112. The hybrid power mean involving the Kloosterman sums and Dedekind sums
  113. Numerical Methods
  114. A modified predictor–corrector scheme with graded mesh for numerical solutions of nonlinear Ψ-caputo fractional-order systems
  115. A kind of univariate improved Shepard-Euler operators
  116. Probability and Statistics
  117. Statistical inference and data analysis of the record-based transmuted Burr X model
  118. Multiple G-Stratonovich integral in G-expectation space
  119. p-variation and Chung's LIL of sub-bifractional Brownian motion and applications
  120. Real Analysis
  121. Chebyshev polynomials of the first kind and the univariate Lommel function: Integral representations
  122. Multiple solutions for a class of fourth-order elliptic equations with critical growth
  123. Majorization-type inequalities for (m, M, ψ)-convex functions with applications
  124. The evaluation of a definite integral by the method of brackets illustrating its flexibility
  125. Some new Fejér type inequalities for (h, g; α - m)-convex functions
  126. Some new Hermite-Hadamard type inequalities for product of strongly h-convex functions on ellipsoids and balls
  127. Topology
  128. Unraveling chaos: A topological analysis of simplicial homology groups and their foldings
  129. A generalized fixed-point theorem for set-valued mappings in b-metric spaces
  130. On SI2-convergence in T0-spaces
  131. Generalized quandle polynomials and their applications to stuquandles, stuck links, and RNA folding
https://doi.org/10.1515/math-2025-0218
Keywords for this article
fuzzy metric spaces; I -semi-α convergence; I -semi-exhaustiveness; I -semi uniform convergence; nets of functions
Creative Commons
BY 4.0

Articles in the same Issue

  1. On I-convergence of nets of functions in fuzzy metric spaces
  2. Special Issue on Contemporary Developments in Graphs Defined on Algebraic Structures
  3. Forbidden subgraphs of TI-power graphs of finite groups
  4. Finite group with some c#-normal and S-quasinormally embedded subgroups
  5. Classifying cubic symmetric graphs of order 88p and 88p 2
  6. Two-sided zero-divisor graphs of orientation-preserving and order-decreasing transformation semigroups
  7. Simplicial complexes defined on groups
  8. Further results on permanents of Laplacian matrices of trees
  9. Algebra
  10. Classes of modules closed under projective covers
  11. On the dimension of the algebraic sum of subspaces
  12. Green's graphs of a semigroup
  13. On an uncertainty principle for small index subgroups of finite fields
  14. On a generalization of I-regularity
  15. Algorithm and linear convergence of the H-spectral radius of weakly irreducible quasi-positive tensors
  16. The hyperbolic CS decomposition of tensors based on the C-product
  17. On weakly classical 1-absorbing prime submodules
  18. Equational characterizations for some subclasses of domains
  19. Algebraic Geometry
  20. Spin(8, ℂ)-Higgs bundles fixed points through spectral data
  21. Embedding of lattices and K3-covers of an Enriques surface
  22. Kodaira-Spencer maps for elliptic orbispheres as isomorphisms of Frobenius algebras
  23. Applications in Computer and Information Sciences
  24. Dynamics of particulate emissions in the presence of autonomous vehicles
  25. Exploring homotopy with hyperspherical tracking to find complex roots with application to electrical circuits
  26. Category Theory
  27. The higher mapping cone axiom
  28. Combinatorics and Graph Theory
  29. 𝕮-inverse of graphs and mixed graphs
  30. On the spectral radius and energy of the degree distance matrix of a connected graph
  31. Some new bounds on resolvent energy of a graph
  32. Coloring the vertices of a graph with mutual-visibility property
  33. Ring graph induced by a ring endomorphism
  34. A note on the edge general position number of cactus graphs
  35. Complex Analysis
  36. Some results on value distribution concerning Hayman's alternative
  37. Freely quasiconformal and locally weakly quasisymmetric mappings in metric spaces
  38. A new result for entire functions and their shifts with two shared values
  39. On a subclass of multivalent functions defined by generalized multiplier transformation
  40. Singular direction of meromorphic functions with finite logarithmic order
  41. Growth theorems and coefficient bounds for g-starlike mappings of complex order λ
  42. Refinements of inequalities on extremal problems of polynomials
  43. Control Theory and Optimization
  44. Averaging method in optimal control problems for integro-differential equations
  45. On superstability of derivations in Banach algebras
  46. The robust isolated calmness of spectral norm regularized convex matrix optimization problems
  47. Observability on the classes of non-nilpotent solvable three-dimensional Lie groups
  48. Differential Equations
  49. The ill-posedness of the (non-)periodic traveling wave solution for the deformed continuous Heisenberg spin equation
  50. A note on the global existence and boundedness of an N-dimensional parabolic-elliptic predator-prey system with indirect pursuit-evasion interaction
  51. Blow-up of solutions for Euler-Bernoulli equation with nonlinear time delay
  52. Periodic or homoclinic orbit bifurcated from a heteroclinic loop for high-dimensional systems
  53. Regularity of weak solutions to the 3D stationary tropical climate model
  54. Local minimizers for the NLS equation with localized nonlinearity on noncompact metric graphs
  55. Global existence and blow-up of solutions to pseudo-parabolic equation for Baouendi-Grushin operator
  56. Bubbles clustered inside for almost-critical problems
  57. Existence and multiplicity of positive solutions for multiparameter periodic systems
  58. Existence of positive periodic solutions for evolution equations with delay in ordered Banach spaces
  59. On a nonlinear boundary value problems with impulse action
  60. Normalized ground-states for the Sobolev critical Kirchhoff equation with at least mass critical growth
  61. Multiple positive solutions to a p-Kirchhoff equation with logarithmic terms and concave terms
  62. Infinitely many solutions for a class of Kirchhoff-type equations
  63. Real and non-real eigenvalues of regular indefinite Sturm–Liouville problems
  64. Existence of global solutions to a semilinear thermoelastic system in three dimensions
  65. Limiting profile of positive solutions to heterogeneous elliptic BVPs with nonlinear flux decaying to negative infinity on a portion of the boundary
  66. Morse index of circular solutions for repulsive central force problems on surfaces
  67. Differential Geometry
  68. On tangent bundles of Walker four-manifolds
  69. Pedal and negative pedal surfaces of framed curves in the Euclidean 3-space
  70. Discrete Mathematics
  71. Eventually monotonic solutions of the generalized Fibonacci equations
  72. Dynamical Systems Ergodic Theory
  73. Dynamical properties of two-diffusion SIR epidemic model with Markovian switching
  74. A note on weighted measure-theoretic pressure
  75. Pullback attractors for a class of second-order delay evolution equations with dispersive and dissipative terms on unbounded domain
  76. Pullback attractor of the 2D non-autonomous magneto-micropolar fluid equations
  77. Functional Analysis
  78. Spectrum boundary domination of semiregularities in Banach algebras
  79. Approximate multi-Cauchy mappings on certain groupoids
  80. Investigating the modified UO-iteration process in Banach spaces by a digraph
  81. Tilings, sub-tilings, and spectral sets on p-adic space
  82. Continuity and essential norm of operators defined by infinite tridiagonal matrices in weighted Orlicz and l∞ spaces
  83. A family of commuting contraction semigroups on l 1 ( N ) and l ∞ ( N )
  84. q-Stirling sequence spaces associated with q-Bell numbers
  85. Chlodowsky variant of Bernstein-type operators on the domain
  86. Hyponormality on a weighted Bergman space of an annulus with a general harmonic symbol
  87. Characterization of derivations on strongly double triangle subspace lattice algebras by local actions
  88. Fixed point approaches to the stability of Jensen’s functional equation
  89. Geometry
  90. The regularity of solutions to the Lp Gauss image problem
  91. Solving the quartic by conics
  92. Group Theory
  93. On a question of permutation groups acting on the power set
  94. A characterization of the translational hull of a weakly type B semigroup with E-properties
  95. Harmonic Analysis
  96. Eigenfunctions on an infinite Schrödinger network
  97. Maximal function and generalized fractional integral operators on the weighted Orlicz-Lorentz-Morrey spaces
  98. Subharmonic functions and associated measures in ℝn
  99. Mathematical Logic, Model Theory and Foundation
  100. A topology related to implication and upsets on a bounded BCK-algebra
  101. Boundedness of fractional sublinear operators on weighted grand Herz-Morrey spaces with variable exponents
  102. Number Theory
  103. Fibonacci vector and matrix p-norms
  104. Recurrence for probabilistic extension of Dowling polynomials
  105. Carmichael numbers composed of Piatetski-Shapiro primes in Beatty sequences
  106. The number of rational points of some classes of algebraic varieties over finite fields
  107. Classification and irreducibility of a class of integer polynomials
  108. Decompositions of the extended Selberg class functions
  109. Joint approximation of analytic functions by the shifts of Hurwitz zeta-functions in short intervals
  110. Fibonacci Cartan and Lucas Cartan numbers
  111. Recurrence relations satisfied by some arithmetic groups
  112. The hybrid power mean involving the Kloosterman sums and Dedekind sums
  113. Numerical Methods
  114. A modified predictor–corrector scheme with graded mesh for numerical solutions of nonlinear Ψ-caputo fractional-order systems
  115. A kind of univariate improved Shepard-Euler operators
  116. Probability and Statistics
  117. Statistical inference and data analysis of the record-based transmuted Burr X model
  118. Multiple G-Stratonovich integral in G-expectation space
  119. p-variation and Chung's LIL of sub-bifractional Brownian motion and applications
  120. Real Analysis
  121. Chebyshev polynomials of the first kind and the univariate Lommel function: Integral representations
  122. Multiple solutions for a class of fourth-order elliptic equations with critical growth
  123. Majorization-type inequalities for (m, M, ψ)-convex functions with applications
  124. The evaluation of a definite integral by the method of brackets illustrating its flexibility
  125. Some new Fejér type inequalities for (h, g; α - m)-convex functions
  126. Some new Hermite-Hadamard type inequalities for product of strongly h-convex functions on ellipsoids and balls
  127. Topology
  128. Unraveling chaos: A topological analysis of simplicial homology groups and their foldings
  129. A generalized fixed-point theorem for set-valued mappings in b-metric spaces
  130. On SI2-convergence in T0-spaces
  131. Generalized quandle polynomials and their applications to stuquandles, stuck links, and RNA folding
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