ℐ-sn-metrizable spaces and the images of semi-metric spaces

Xiangeng Zhou; Fang Liu; Li Liu; Shou Lin

doi:10.1515/math-2024-0053

Article Open Access

ℐ-sn-metrizable spaces and the images of semi-metric spaces

Xiangeng Zhou , Fang Liu , Li Liu and Shou Lin

Published/Copyright: August 23, 2024

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$Open Mathematics$

From the journal Open Mathematics Volume 22 Issue 1

Abstract

The theory of generalized metric spaces is an active topic in general topology. In this article, we utilize the concepts of ideal convergence and networks to discuss the metrization problem and the mutual classification problem between spaces and mappings in topological spaces. We define ℐ - s n -metrizable spaces, obtain several characterizations of ℐ - s n -metrizable spaces, and establish some mapping relations between ℐ - s n -metrizable spaces and semi-metric spaces. These not only generalize some theorems in generalized metric theory, but also find further applications of ideal convergence in general topology.

Keywords: ideal convergence; ℐ-sn-networks; ℐ-cs-networks; ℐ-sn-metrizable spaces; semi-metric spaces; ℵ-spaces; ℐ-covering mappings; π-mappings; σ-mappings

MSC 2010: 54A20; 54C10; 54E35; 54E40; 54E99

1 Introduction

The theory of generalized metric spaces is an important research topic in general topology [1]. Its origin and development mainly stem from the following three problems: the metrization problem, the problem of paracompactness in product spaces, and the mutual classification problem between spaces and mappings [2]. There are a number of reasons why generalized metric spaces are worthy of study. Perhaps the most important reason is that such classes increase our understanding of metrizable spaces. In addition, topologists are continually seeking broader classes of spaces for which especially important results hold [3].

g -metrizable spaces, s n -metrizable spaces, and ℵ -spaces are all important classes of generalized metric spaces. Over the past 40 years, extensive research has been conducted on them; thereby, their position in the theory of generalized metric spaces is established [1]. The importance of g -metrizable spaces is not only because it is a generalized metric space based on the concept of weak bases introduced by Arhangel’skiĭ [4], but also because it has beautiful characterizations similar to the classical metrization theorems [3].

Theorem 1.1

[1, Theorems 3.9.3 and 3.9.11] The following are equivalent for a regular space X:

X is a g-metrizable space.
X has a σ -discrete weak base.
X has a σ -hereditarily closure-preserving weak base.
X has a point-star weak base consisting of locally finite covers.

It is known that every g -metrizable space is an s n -metrizable space, and every s n -metrizable space is an ℵ -space [5, Remark 1.3]. It can be seen that s n -metrizable spaces play a bridging role between g -metrizable spaces and ℵ -spaces. More importantly, s n -metrizable spaces also have nice properties.

Theorem 1.2

[6, Theorem 1] The following are equivalent for a regular space X:

X is an sn-metrizable space.
X has a σ -discrete sn-network.
X has a σ -hereditarily closure-preserving sn-network.
X has a point-star sn-network consisting of locally finite covers [7, Theorem 2.5].
X is an snf-countable and ℵ -space.

In terms of the mutual classification between spaces and mappings, it is known that thre exists a g -metrizable space which is not a sequence-covering, π and σ -image of a metric space [8, Example 3.1]. With the help of s n -symmetric spaces (or semi-metric spaces) and sequence-covering mappings, the following mapping theorem of s n -metrizable spaces is established.

Theorem 1.3

[8, Theorem 3.5] The following are equivalent for a regular space X:

X is an sn-metrizable space.
X is a 1-sequence-covering, π and σ -image of a semi-metric space.
X is a sequence-covering, π and σ -image of an sn-symmetric space.

The key concept of Theorems 1.2 and 1.3 is s n -networks, which were introduced by Lin [9] as a natural generalization of weak bases. In recent years, research on s n -networks became an active field [1,10–12].

The essence of an s n -network is a network determined by convergent sequences, i.e., a network of sequential neighborhoods. Convergence of sequences in a topological space is a basic and important concept in mathematics [1,13]. In addition to the usual convergence of sequences, statistical convergence [14–16] and ideal convergence [17–21] have attracted extensive attention. In particular, ideals are a very useful notion in topology, analysis, and set theory and have been studied for a long time. In [22–25], certain topological spaces and networks defined by ideal convergence were studied.

Meanwhile, ideal convergence [17,18,26] and networks that are defined by ideal convergence have become active. Specifically, ℐ - s n -networks and ℐ - c s -networks were introduced in [23,24], which deepened the research studies on the spaces with ideal convergence. As we know, mappings are an important tool to study spaces and they play a pivotal role in discussing various images of metric spaces [1,4]. Sequence-covering mappings are a special kind of mappings [27]. Through sequence-covering mappings, we can establish the relationship between convergence sequences in topological spaces, and further reveal some topological properties based on convergent sequences [10]. For example, we introduced and discussed more general sequence-covering mappings, that is, ℐ -covering mappings. Through ℐ -covering mappings, we study the properties of ℐ -convergent sequences in topological spaces and investigate seq-type, FU-type, and ℐ - s n f -properties in topological spaces [21,23,25,28,29]. As some extensions of Theorems 1.2 and 1.3, we can attempt to define ℐ - s n -metrizable spaces and study the following problems.

Problem 1.4

Find a characterization of ℐ - s n -metrizable spaces by certain families of subsets analogous to Theorem 1.2.

Problem 1.5

Find a characterization of ℐ - s n -metrizable spaces by certain ℐ -covering images of semi-metric spaces analogous to Theorem 1.3.

The main goal of this article is to innovate ℐ - s n -metrizable spaces and answer Problems 1.4 and 1.5 (Theorems 3.3, 4.6, and 4.10). The article is organized as follows. In Section 2, the necessary notation and terminology are introduced; in particular, ℐ - s n -metrizable spaces are defined; and some basic relations among the spaces defined by these notions are described. In Section 3, we give some relationship among ℐ - s n -metrizable spaces, ℐ - s n f -countable spaces, and ℵ -spaces and obtain several characterizations of ℐ - s n -metrizable spaces, which give an answer to Problem 1.4. In Section 4, using the notion of point-star ℐ - s n -networks, we characterize the images of semi-metric spaces under ℐ -covering, π and σ -mappings, which answers Problem 1.5.

We believe that the discussion on this topic has at least the following significance. First, these results generalize some theorems in s n -metrizable spaces, g -metrizable spaces, and semi-metric spaces [5–8]. Second, we draw into ℐ - s n -metrizable spaces and ℐ - s n -symmetric spaces, which bring ideal convergence into generalized metric spaces, and greatly enrich the applications of ideal convergence. Third, these results prompt the study of the relationship between the spaces defined by ideal convergence and mappings, which further improves the mutual classification between spaces and mappings.

In the following, if not otherwise specified, we consider all spaces are assumed to be T 1 and all mappings are continuous and surjection.

2 ℐ - s n -networks and related results

In this section, we recall the necessary notation, terminology, and describe some basic relations among spaces defined by these notions. For simplicity, sometimes the notation “ A ⇒ B ” (resp., “ A ⇔ B ”) is used to represent “ A implies B ” (resp., “ A is equivalent to B ”). The readers may refer to [1,13] for notation and terminology not explicitly given here.

The concept of ℐ -convergence in topological spaces is a generalization of statistical convergence and the usual convergence, which is based on the ideal of subsets of the set N of all positive integers. Let A = 2 N be the family of all subsets of N . An ideal ℐ ⊆ A is a hereditary family of subsets of N which is stable under finite unions [26, P. 670], i.e., the following are satisfied: if B ⊆ A ∈ ℐ , then B ∈ ℐ ; if A , B ∈ ℐ , then A ∪ B ∈ ℐ . An ideal ℐ is said to be n o n - t r i v i a l , if ℐ ≠ ∅ and N ∉ ℐ . A non-trivial ideal ℐ ⊆ A is called a d m i s s i b l e if ℐ ⊇ { { n } : n ∈ N } . The family of all finite subsets of N is denoted by ℐ fin , which is the smallest ideal contained in each admissible ideal. In this article, we consider ℐ is always an admissible ideal on N .

Recall that a sequence { x n } n ∈ N in a topological space X is said to be ℐ - c o n v e r g e n t to a point x ∈ X , provided that for any neighborhood U of x , we have { n ∈ N : x n ∉ U } ∈ ℐ , which is denoted by x n → ℐ x [26, Definition 3.1].

Let P ⊆ X . A sequence { x n } n ∈ N in X is said to be ℐ -eventually in P if the set { n ∈ N : x n ∉ P } ∈ ℐ [25, Definition 3.15]. The set P is said to be an ℐ -sequential neighborhood of a point x ∈ X if each sequence which is ℐ -convergent to x is ℐ -eventually in P [21, Definition 2.5]. The set P is said to be an ℐ s n -open set of X if P is an ℐ -sequential neighborhood of x for each x ∈ P [22, P. 4]. ℐ fin -sequential neighborhood of x or ( ℐ fin ) s n -open sets are called sequential neighborhood of x or sequentially open sets, respectively [30, P. 108].

Next, we will introduce several interior operators in topological spaces formed by ℐ -convergence [22]. Let X be a topological space and A ⊆ X . Put

( A ) seq = { x ∈ X : A is a sequential neighborhood of x } , ( A ) ℐ s n = { x ∈ X : A is an ℐ -sequential neighborhood of x } .

The following lemma illustrates the relationship between several forms of open sets and several operations, which is the foundation for studying ideal topological spaces.

Lemma 2.1

Let X be a topological space and A ⊆ X .

If A is open in X, then A is ℐ s n -open; if A is ℐ s n -open in X, then A is sequentially open [22, Lemma 2.1].
A ∘ ⊆ ( A ) ℐ s n ⊆ ( A ) seq ⊆ A [22, Lemma 2.6].

Definition 2.2

Let X be a topological space.

X is called an ℐ s n -sequential space [24, Definition 3.1] (resp., sequential space [13, P. 53]), provided that any ℐ s n -open (resp., sequentially open) subset of X is open.
X is called an ℐ s n -FU-space [24, Definition 3.1] (resp., FU- ℐ s n -space [28, Definition 2.2]), provided ( A ) ℐ s n ⊆ A ∘ (resp., ( A ) seq ⊆ ( A ) ℐ s n ) for each A ⊆ X .

By Definition 2.2, it is easy to see that every sequential space or ℐ s n -FU-space is an ℐ s n -sequential space.

The concept of networks has played a key role in the study of topological spaces. Let P be a family of subsets of a topological space X . The family P is a network at some point x in X if x ∈ ⋂ P and whenever x ∈ U with U open in X , then P ⊆ U for some P ∈ P [13, P. 127]. P is said to be discrete [1, Definition 1.2.2] (resp., locally finite [1, Definition 1.3.1]) in X if, for each x ∈ X , there is a neighborhood V of x in X such that V meet at most one element (resp., finitely many elements) of P . The union of countably many discrete (resp., locally finite) families is called a σ -discrete (resp., σ -locally finite) family.

Definition 2.3

Let P be a family of subsets of a topological space X .

P is called an ℐ - c s -network at a point x ∈ X if whenever { x n } n ∈ N is a sequence ℐ -converging to x ∈ U with U open in X , then { x } ∪ { x n : n ∈ N \ I } ⊆ P ⊆ U for some I ∈ ℐ and P ∈ P [29, Definition 4.1].
P is called an ℐ - c s -network of X if P is an ℐ - c s -network at each point x ∈ X [29, Definition 4.1]. Every ℐ fin -cs-network is called a c s -network [31, P. 106].
X is called an ℵ -space if X is a regular space with a σ -locally finite c s -network [10, P. 8].

It is easy to check that the statement “ { x n : n ∈ N \ I } ⊆ P for some I ∈ ℐ ” is equivalent to the statement “ { n ∈ N : x n ∉ P } ∈ ℐ ” in Definition 2.3, i.e., the sequence { x n } n ∈ N is ℐ -eventually in the set P .

Definition 2.4

Let P = ⋃ x ∈ X P x be a family of subsets of a topological space X satisfying the following (a) and (b) for every x ∈ X .

x ∈ ⋂ P x ;
if U , V ∈ P x , then W ⊆ U ∩ V for some W ∈ P x .

(1) P is called a weak base of X , provided that G ⊆ X is open if and only if for each x ∈ G there is P ∈ P x such that P ⊆ G [4]. The family P x is called a weak base at x in X .

X is called a g f -countable space, provided that X has a countable weak base at each point in X [4]. X is called a g -metrizable space, provided that X is a regular space with a σ -locally finite weak base [5, Definition 1.3].

(2) P is called an ℐ - s n -network of X , provided that for each x ∈ X , P x is a network at x in X consisting of ℐ -sequential neighborhoods of x . The family P x is called an ℐ - s n -network at x in X [24, Definition 5.1].

X is called an ℐ - s n f -countable space if X has a countable ℐ - s n -network at each point in X [24, Definition 4.1]. X is called an ℐ - s n -metrizable space, provided that X is a regular space with a σ -locally finite ℐ - s n -network.

The ℐ - s n -metrizable space in Definition 2.4 is a new concept defined and focused on in this article. An ℐ fin - s n -network is called an s n -network [9, Definition 2.7]; an ℐ fin - s n f -countable space is called an s n f -countable space [9, Definition 2.7]; an ℐ fin - s n -metrizable space is called an s n -metrizable space [5, Definition 1.3].

Let P be a cover of a topological space X . P is called a c s ∗ -network of X if whenever a sequence { x n } n ∈ N converges to a point x ∈ U with U open in X , then there is a subsequence { x n i } i ∈ N of { x n } n ∈ N such that { x } ∪ { x n i : i ∈ N } ⊆ P ⊆ U for some P ∈ P [10, P. 8].

Remark 2.5

For a topological space, bases ⇒ ℐ - s n -networks ⇒ ℐ - c s -networks and s n -networks [24, Lemma 5.2]; and c s -networks ⇒ c s ∗ -networks.
X is an ℵ -space ⇔ X is a regular space with a σ -locally finite c s ∗ -network ⇔ X is a regular space with a σ -discrete c s -network [32, Theorem 4].
An ℐ - s n f -countable space X is ℐ s n -sequential if and only if it is sequential [24, Corollary 4.6].
g -metrizable spaces ⇔ sequential and s n -metrizable spaces [8, Remark 2.7].

3 ℐ - s n -metrizable spaces

In this section, we give some relationship among ℐ - s n -metrizable spaces, ℐ - s n f -countable spaces, and ℵ -spaces and obtain some characterizations of ℐ - s n -metrizable spaces. These results generalize some classic conclusions on s n -metrizable spaces, g -metrizable spaces, and metrizable spaces.

The following example shows that an ℐ - s n -metrizable space is a non-trivial generalization of an s n -metrizable space.

Example 3.1

There is an s n -metrizable space which is not an ℐ - s n -metrizable space.

Proof

Let ℐ be an ideal on N . Define a topology τ on the set N ∪ { ∞ } , ∞ ∉ N , as follows:

Each point n ∈ N is isolated.
Each open neighborhood U of ∞ is of the form ( N \ I ) ∪ { ∞ } , for each I ∈ ℐ .

We denote the set N ∪ { ∞ } equipped with this topology by Σ ( ℐ ) . Since { n ∈ N : n ∉ ( N \ I ) ∪ { ∞ } } = I ∈ ℐ for each I ∈ ℐ , the sequence n → ℐ ∞ in Σ ( ℐ ) .

Now, we consider that ℐ is some maximal ideal on N [33, Exercise 4M].

It is easy to see that there is an open and closed neighborhood base for each point in Σ ( ℐ ) ; hence, X is regular.
Since there is not any non-trivial convergent sequence in Σ ( ℐ ) [33, Exercise 4M], P = { { x } : x ∈ Σ ( ℐ ) } is a countable s n -network of Σ ( ℐ ) . Hence, Σ ( ℐ ) is an s n -metrizable space.
However Σ ( ℐ ) is not an ℐ - s n f -countable space [24, Example 5.8]. Thus, Σ ( ℐ ) is not an ℐ - s n -metrizable space.□

Lemma 3.2

Every ℐ - c s -network of a topological space X is a c s ∗ -network.

Proof

Suppose that P is an ℐ - c s -network of X . Let { x n } n ∈ N be a sequence in X such that x n → x and U be an open subset of X with x ∈ U . Thus, x n → ℐ x . Since P is an ℐ - c s -network of X , there exist P ∈ P and I ∈ ℐ such that { x } ∪ { x n : n ∈ N \ I } ⊆ P ⊆ U . Denote { x n : n ∈ N \ I } by { x n i } i ∈ N . Then { x } ∪ { x n i : i ∈ N } ⊆ P . This implies that P is a c s ∗ -network of X .□

The following is a main result in this section, which gives a partial answer to Problem 1.4. It illustrates the relationship among ℐ - s n -metrizable spaces, ℐ - s n f -countable spaces, and ℵ -spaces; and it lays the foundation for establishing more properties of the spaces defined by ideal convergence.

Theorem 3.3

The following are equivalent for a regular space X:

X is an ℐ -sn-metrizable space.
X is an ℐ -snf-countable space with a σ -locally finite ℐ -cs-network.
X is an ℐ -snf-countable space with a σ -locally finite c s ∗ -network.
X is an ℐ -snf-countable and ℵ -space.
X has a σ -discrete ℐ -sn-network.

Proof

It is obvious that every ℐ - s n -metrizable space is an s n -metrizable, ℐ - s n f -countable, and ℵ -space. It follows from parts (1) and (2) in Remark 2.5 and Lemma 3.2 that (1) ⇒ (2) ⇒ (3) ⇒ (4) and (5) ⇒ (1). We only need to prove that (4) ⇒ (5).

(4) ⇒ (5). Since X is an ℵ -space, it follows from part (2) in Remark 2.5 that X has a σ -discrete c s -network P = ⋃ n ∈ N P n , where each P n is discrete and P is closed under finite intersections. For each x ∈ X , let Q x be a countable and decreasing ℐ - s n -network at x in X , and put

P x ′ = { P ∈ P : Q ⊆ P for some Q ∈ Q x } .

It is clear that each element of P x ′ is an ℐ -sequential neighborhood of x and for any P ′ , P ″ ∈ P x ′ , we have P ′ ∩ P ″ ∈ P x ′ .

(i) P x ′ is a network at x in X . Note that P is σ -discrete, then P is point-countable. Let Q x = { Q n ( x ) : n ∈ N } with Q n + 1 ( x ) ⊆ Q n ( x ) and U be an open subset of X with x ∈ U . Then, there exists P ∈ P x ′ such that P ⊆ U . Otherwise, let

{ P ∈ P : x ∈ P ⊆ U } = { P m ( x ) : m ∈ N } .

Thus, Q n ( x ) ⊈ P m ( x ) for each n , m ∈ N . Choose x n m ∈ Q n ( x ) \ P m ( x ) . For n ≥ m , let y k = x n m , where k = m + n ( n − 1 ) 2 . Since { Q n ( x ) : n ∈ N } is decreasing, it follows that the sequence { y k } k ∈ N converges to the point x . Thus, there are m , i ∈ N such that { y k : k ≥ i } ∪ { x } ⊆ P m ( x ) ⊆ U . Take j ≥ i with y j = x n m for some n ≥ m . Then x n m ∈ P m ( x ) . This is a contradiction. Thus, (i) is true.

(ii) P ′ = ⋃ { P x ′ : x ∈ X } is a σ -discrete ℐ - s n -network of X . In fact, since P ′ ⊆ P and P is σ -discrete, it follows that P ′ is a σ -discrete ℐ - s n -network of X .□

The following result provides a metrization theorem for ℐ - s n -metrizable spaces, which is a generalization of a metrization theorem for s n -metrizable spaces [5, Theorem 2.4].

Corollary 3.4

X is a metrizable space if and only if it is an ℐ s n -FU, ℐ - s n -metrizable space.

Proof

It is obvious that every metrizable space is an ℐ s n -FU, ℐ - s n -metrizable space. Now, let X be an ℐ s n -FU, ℐ - s n -metrizable space. It is clear that X is an ℐ - s n f -countable and s n -metrizable space. Since every ℐ s n -FU, ℐ - s n f -countable space is first-countable [24, Corollary 4.8], X is first-countable. Thus, X is a sequential space. By part (4) in Remark 2.5, X is a g -metrizable space. Since every first-countable, g -metrizable space is metrizable [1, Corollary 3.9.4], the space X is metrizable.□

According to Theorem 1.2, it is natural to consider the following problem. A family P of subsets of a topological space X is said to be hereditarily closure-preserving [1, Definition 2.5.2] in X if, for each H ( P ) ⊆ P ∈ P , the formula ⋃ { P ( H ) : P ∈ P } ¯ = ⋃ { P ( H ) ¯ : P ∈ P } holds.

Problem 3.5

Let X be a regular space with a σ -hereditarily closure-preserving ℐ - s n -network. Is X ℐ - s n -metrizable?

The following result establishes the exact relationship between ℐ - s n -metrizable spaces and s n -metrizable spaces, and the difference between them can also be seen from this. It follows from Example 3.1 that the condition of ℐ - s n f -countable or FU- ℐ s n -spaces in Theorem 3.6 is essential.

Theorem 3.6

The following are equivalent for a topological space X:

X is an ℐ -sn-metrizable space.
X is an ℐ -snf-countable and sn-metrizable space.
X is an FU- ℐ s n and sn-metrizable space.

Proof

(1) ⇒ (2). By part (1) in Remark 2.5, it is clear.

(2) ⇒ (3). It suffices to show that every ℐ - s n f -countable space is an FU- ℐ s n -space. In fact, let X be an ℐ - s n f -countable space. Assume that there is a point x ∈ ( A ) seq \ ( A ) ℐ s n for some A ⊆ X . Then, there is a sequence { a n } n ∈ N in X which is ℐ -convergent to x but not ℐ -eventually in A . Since X is an ℐ - s n f -countable space, there is a countable ℐ - s n -network { P m } m ∈ N at x , and so the sequence { a n } n ∈ N is ℐ -eventually in each P m . For each k ∈ N , put Q k = ⋂ m ≤ k P m . If I 1 , I 2 ∈ ℐ , then

( { x } ∪ { a n : n ∈ N \ I 1 } ) ∩ ( { x } ∪ { a n : n ∈ N \ I 2 } ) = { x } ∪ { a n : n ∈ N \ ( I 1 ∪ I 2 ) } .

It follows that the sequence { a n } n ∈ N is still ℐ -eventually in Q k . Thus, the set Q k ⊈ A , and there is x k ∈ Q k \ A . Note that { Q k } k ∈ N is a decreasing network at x in X . Hence, the sequence { x k } k ∈ N is convergent to x . This contradicts to A being a sequential neighborhood of x . Thus, X is an FU- ℐ s n -space.

(3) ⇒ (1). Let X be an FU- ℐ s n and s n -metrizable space. Then, each sequential neighborhood at each point in X is an ℐ -sequential neighborhood at the point. Thus, each s n -network at each point in X is an ℐ - s n -network at the point. Therefore, X is an ℐ - s n -metrizable space.□

Corollary 3.7

The following are equivalent for a topological space X:

X is a sequential and ℐ -sn-metrizable space.
X is an ℐ s n -sequential and ℐ -sn-metrizable space.
X is an ℐ -snf-countable and g-metrizable space.
X is an FU- ℐ s n and g-metrizable space.

Proof

(1) ⇒ (2). It is obvious.

(2) ⇒ (3). Let X be an ℐ s n -sequential and ℐ - s n -metrizable space. Then, X is an ℐ - s n f -countable and s n -metrizable space. By part (3) in Remark 2.5, X is a sequential space. It follows from part (4) in Remark 2.5 that X is a g -metrizable space.

(3) ⇒ (4). By the proof of Theorem 3.6, we see that every ℐ - s n f -countable space is an FU- ℐ s n -space.

(4) ⇒ (1). Let X be an FU- ℐ s n and g -metrizable space. Then, X is an FU- ℐ s n , sequential and s n -metrizable space. It follows from Theorem 3.6 that X is a sequential and ℐ - s n -metrizable space.□

It is necessary to consider the conditions given in Corollary 3.7, and the following examples and problem explain these.

Example 3.8

There is an ℐ - s n -metrizable space which is not an ℐ s n -sequential space.

Proof

Let X = N ∪ { p } , where p ∈ β N \ N and the set X is endowed with the subspace topology τ of the Čech-Stone compactification β N . Taking ℐ = ℐ fin , then X is not a discrete space. Since the space X has no non-trivial convergent sequence [13, Corollary 3.6.15], every subset of X is ( ℐ fin ) s n -sequentially open in X . Thus, ( X , τ ) is not an ( ℐ fin ) s n -sequential space, i.e., X is not a sequential space. It is clear that P = { { x } : x ∈ X } is a countable ℐ fin - s n -network of X . Hence, X is an ℐ fin - s n -metrizable space.□

Example 3.9

There is an ℐ s n -sequential and s n -metrizable space which is neither a g -metrizable space nor an ℐ - s n -metrizable space.

Proof

Let Σ ( ℐ ) be the space in Example 3.1, where ℐ is some maximal ideal on N [33, Exercise 4M].

Since Σ ( ℐ ) is an ℐ s n -FU-space [24, Example 3.4], it is an ℐ s n -sequential space.
It follows from Example 3.1 that Σ ( ℐ ) is an s n -metrizable space which is not an ℐ - s n -metrizable space.
Note that X is not a sequential space [25, Example 3.9]. Hence, X is not a g -metrizable space.□

Problem 3.10

Is every g -metrizable space an ℐ - s n f -countable space for each admissible ideal ℐ ?

4 Images of semi-metric spaces

In this section, we first characterize ℐ - s n -metrizable spaces by point-star ℐ - s n -networks consisting of locally finite covers. Next, we characterize the images of semi-metric spaces ( ℐ - s n -symmetric spaces) under 1- ℐ -covering ( ℐ -covering), π and σ -mappings.

Recall the concept of semi-metric spaces. Let X be a topological space. A real-valued function d : X × X → [ 0 , + ∞ ) is called a d -function if it satisfies the following conditions:

d ( x , y ) = 0 if and only if x = y for each x , y ∈ X ;
d ( x , y ) = d ( y , x ) , for each x , y ∈ X [10, P. 4].

Let d be a d -function on a topological space X . The space ( X , d ) is called a symmetric space [10, P. 4] if d satisfies the condition: a subset U in X is open if and only if for each x ∈ X , there exists n ∈ N such that B ( x , 1 n ) ⊆ U , where B ( x , 1 n ) = { y ∈ X : d ( x , y ) < 1 n } . The space X is called a semi-metric space [10, P. 4] if X is a first-countable symmetric space.

Definition 4.1

Let X be a topological space and d be a d -function on X . The space ( X , d ) is called an ℐ - s n -symmetric space if d satisfies the condition: { B ( x , 1 n ) } n ∈ N is an ℐ - s n -network at x in X for every x ∈ X . Then, d is called an ℐ - s n -symmetric on X . An ℐ fin - s n -symmetric space is called an s n -symmetric space [10, Definition 3.1.2].

The ℐ - s n -symmetric space is another new space defined in this article, and its function can be reflected in Lemma 4.8 and Theorem 4.10.

Proposition 4.2

X is an ℐ -sn-symmetric space if and only if X is an ℐ -snf-countable and sn-symmetric space.

Proof

Necessity. Let X be an ℐ - s n -symmetric space. By Definition 4.1 and part (1) in Remark 2.5, it is obvious that X is an ℐ - s n f -countable and s n -symmetric space.

Sufficiency. Let X be an ℐ - s n f -countable and s n -symmetric space. By the proof of Theorem 3.6, we see that every ℐ - s n f -countable space is an FU- ℐ s n -space. Thus, X is an ℐ - s n -symmetric space.□

Remark 4.3

X is a symmetric space if and only if X is a sequential and s n -symmetric space [8, part (2) in Remark 2.7].
X is a semi-metric space if and only if X is a first-countable and s n -symmetric space [8, part (4) in Remark 2.7].

Definition 4.4

Let X , Y be topological spaces and f : X → Y be a mapping.

f is called a σ -mapping, if there exists a base ℬ of X such that f ( ℬ ) is σ -locally finite in Y [10, P. 147].
f is called an ℐ -covering mapping, if whenever a sequence y n → ℐ y in Y , there is a sequence x n → ℐ x in X satisfying x ∈ f − 1 ( y ) and each x n ∈ f − 1 ( y n ) [25, Definition 5.1].
f is called a 1- ℐ -covering mapping, if for every y ∈ Y , there exists x ∈ f − 1 ( y ) such that whenever { y n } n ∈ N is a sequence ℐ -converging to y in Y , there exists a sequence { x n } n ∈ N ℐ -converging to x in X with each x n ∈ f − 1 ( y n ) [21, Definition 4.6].
Suppose that ( X , d ) is a metric (resp., ℐ - s n -symmetric, or semi-metric) space. f is called a π -mapping with respect to d , if d ( f − 1 ( y ) , X \ f − 1 ( U ) ) > 0 for every y ∈ Y and every neighborhood U of y in Y [10, Definition 3.2.1].

Let P be a family of subsets of X . For each x ∈ X , denote ( P ) x = { P ∈ P : x ∈ P } , and st ( x , P ) = ⋃ ( P ) x . The following concept of point-star networks is an important bridge to establish the connection between Problems 1.4 and 1.5, see Theorems 4.6 and 4.10.

Definition 4.5

Let { P n } n ∈ N be a sequence of covers of a topological space X .

{ P n } n ∈ N is said to be a point-star network of X , if { st ( x , P n ) } n ∈ N is a network at x for every x ∈ X [10, Definition 3.1.1].
{ P n } n ∈ N is said to be a point-star ℐ - s n -network of X , if { st ( x , P n ) } n ∈ N is an ℐ - s n -network at x for every x ∈ X .

Obviously, { P n } n ∈ N is a point-star network of X if and only if for each x ∈ X and P n ∈ ( P n ) x , then the family { P n } n ∈ N is a network at x in X [10, P. 122].

Let P and Q be families of subsets of X . Put P ∧ Q = { P ∩ Q : P ∈ P , Q ∈ Q } . The following theorem gives a partial answer to Problem 1.4.

Theorem 4.6

The following are equivalent for a topological space X:

X is an ℐ -sn-metrizable space.
X is a regular space with a point-star ℐ -sn-network consisting of locally finite covers.

Proof

(1) ⇒ (2). Let X be an ℐ - s n -metrizable space. It follows from the regularity of X and Theorem 3.3 that X has an ℐ - s n -network P = ⋃ { P n : n ∈ N } , where each P n is a discrete collection of closed subsets of X . Denote P = ⋃ { ℬ x : x ∈ X } , where each ℬ x is an ℐ - s n -network at x . For each n ∈ N , put F n = { x ∈ X : P n ∩ ℬ x = ∅ } , ℱ n = { F n } ∪ P n . Obviously, { ℱ n } n ∈ N is a sequence of locally finite covers of X . For each x ∈ X , let ℱ x = { st ( x , ℱ n ) } n ∈ N .

ℱ x is a network at x . Let U be an open neighborhood of x . Since ℬ x is a network at x , there exists P x ∈ ℬ x ∩ P n such that x ∈ P x ⊆ U for some n ∈ N . Thus, x ∉ F n . Note that elements of P n are mutually disjoint. So, P x = st ( x , ℱ n ) , that is, x ∈ st ( x , ℱ n ) ⊆ U .
st ( x , ℱ n ) is an ℐ -sequential neighborhood of x for each n ∈ N . Let L be a sequence ℐ -converging to x ∈ st ( x , ℱ n ) . If ℬ x ∩ P n ≠ ∅ , then there exists P ∈ ℬ x ∩ P n . Since P is an ℐ -sequential neighborhood of x , L is ℐ -eventually in P , thus L is ℐ -eventually in st ( x , ℱ n ) . If ℬ x ∩ P n = ∅ , put U = X \ ⋃ { P ∈ P n : x ∉ P } . Then U is an open neighborhood of x , hence L is ℐ -eventually in U . It is easy to see that U ⊆ st ( x , ℱ n ) , so L is ℐ -eventually in st ( x , ℱ n ) . This implies that st ( x , ℱ n ) is an ℐ -sequential neighborhood of x for each n ∈ N .

It is obvious that for each n ∈ N , the family ⋀ i ≤ n ℱ i is a locally finite cover in X . Note that st ( x , ⋀ i ≤ n ℱ i ) = ⋂ i ≤ n st ( x , ℱ i ) , hence { st ( x , ⋀ i ≤ n ℱ i ) } n ∈ N is an ℐ - s n -network at x . This proves that { ⋀ i ≤ n ℱ i } n ∈ N is a point-star ℐ - s n -network consisting of locally finite covers.

(2) ⇒ (1). Suppose that X has a point-star ℐ - s n -network { P n } n ∈ N consisting of locally finite covers. It is easy to see that X is an ℐ - s n f -countable space. Next, we will show that ⋃ n ∈ N P n is a σ -locally finite c s ∗ -network of X . In fact, let { x k } k ∈ N be a sequence in X and x k → x ∈ U with U open in X . Thus x k → ℐ x . Since { P n } n ∈ N is a point-star ℐ - s n -network, there are n ∈ N and I ∈ ℐ such that { x } ∪ { x k : k ∈ N \ I } ⊆ st ( x , P n ) ⊆ U . Note that P n is point-finite, so there exists a subsequence { x k i } i ∈ N of { x k } k ∈ N \ I such that { x } ∪ { x k i : i ∈ N } ⊆ P ⊆ st ( x , P n ) ⊆ U for some P ∈ P n . This implies that ⋃ n ∈ N P n is a σ -locally finite c s ∗ -network of X . By Theorem 3.3, X is an ℐ - s n -metrizable space.□

Corollary 4.7

Every ℐ -sn-metrizable space is an ℐ -sn-symmetric space.

Proof

Let X be an ℐ - s n -metrizable space. It follows from Theorem 4.6 that X has a point-star ℐ - s n -network { P n } n ∈ N . Define a function d : X × X → [ 0 , + ∞ ) as follows: for each x , y ∈ X ,

d ( x , y ) = 0 , y ∈ ⋂ n ∈ N st ( x , P n ) , sup { 1 ⁄ n : y ∉ st ( x , P n ) } , y ∉ ⋂ n ∈ N st ( x , P n ) .

It is obvious that d ( x , y ) = d ( y , x ) , and d ( x , y ) = 0 if and only if x = y , because X is a T 1 space. If n ∈ N , then y ∉ st ( x , P n ) if and only if d ( x , y ) ≥ 1 ⁄ n if and only if y ∉ B ( x , 1 ⁄ n ) , thus st ( x , P n ) = B ( x , 1 ⁄ n ) . This implies that ( X , d ) is an ℐ - s n -symmetric space.□

Next, we will obtain some intrinsic properties of the images of semi-metric (resp., ℐ - s n -symmetric) spaces under 1- ℐ -covering (resp., ℐ -covering), π and σ -mappings.

Lemma 4.8

Let M be an ℐ - s n -symmetric space and f : M → X be a mapping.

If X is an ℐ - s n -symmetric space, then f is a π -mapping with respect to some ℐ - s n -symmetric on M.
If f is an ℐ -covering, π -mapping, then X is an ℐ - s n -symmetric space.

Proof

(1) Let d , d ′ be ℐ - s n -symmetries on spaces M and X , respectively. Put

δ ( a , b ) = d ( a , b ) + d ′ ( f ( a ) , f ( b ) ) , for each a , b ∈ M .

It is clear that δ is a d -function on M . Let a ∈ M , x ∈ X and n ∈ N . Suppose that

B δ , n ( a ) = b ∈ M : δ ( a , b ) < 1 n , B d , n ( a ) = b ∈ M : d ( a , b ) < 1 n , B d ′ , n ( x ) = y ∈ X : d ′ ( x , y ) < 1 n .

(i) { B δ , n ( a ) } n ∈ N is a network at a in M for each a ∈ M .

Let a ∈ U with U open in M . Since d is an ℐ - s n -symmetric on M , there exists n ∈ N such that B d , n ( a ) ⊆ U . Note that d ( a , b ) ≤ δ ( a , b ) for each b ∈ M , then B δ , n ( a ) ⊆ B d , n ( a ) ⊆ U . Hence, { B δ , n ( a ) } n ∈ N is a network at a in M .

(ii) B δ , n ( a ) is an ℐ -sequential neighborhood of a for each a ∈ M and n ∈ N .

Let { a k } k ∈ N be a sequence in M with a k → ℐ a . Then, f ( a k ) → ℐ f ( a ) . Note that { B d , n ( a ) } n ∈ N is an ℐ - s n -network at a in M and { B d ′ , n ( f ( a ) ) } n ∈ N is an ℐ - s n -network at f ( a ) in X . Thus, there are I 1 , I 2 ∈ ℐ such that d ( a , a k ) < 1 2 n for k ∈ N \ I 1 and d ′ ( f ( a ) , f ( a k ) ) < 1 2 n for k ∈ N \ I 2 . Hence, for each k ∈ N \ ( I 1 ∪ I 2 ) ,

δ ( a , a k ) = d ( a , a k ) + d ′ ( f ( a ) , f ( a k ) ) < 1 2 n + 1 2 n = 1 n .

That is a k ∈ B δ , n ( a ) for k ∈ N \ ( I 1 ∪ I 2 ) . Since I 1 ∪ I 2 ∈ ℐ , it follows that the sequence { a k } k ∈ N is ℐ -eventually in B δ , n ( a ) . Therefore, B δ , n ( a ) is an ℐ -sequential neighborhood of a in M .

In view of (i) and (ii), δ is an ℐ - s n -symmetric on M .

(iii) f is a π -mapping with respect to δ .

Let x ∈ U with U open in X . There exists n ∈ N such that B d ′ , n ( x ) ⊆ U . If a ∈ f − 1 ( x ) and b ∈ M \ f − 1 ( U ) , then f ( b ) ∉ U and d ′ ( x , f ( b ) ) ≥ 1 n . Thus

δ ( a , b ) ≥ d ′ ( f ( a ) , f ( b ) ) = d ′ ( x , f ( b ) ) ≥ 1 n .

So δ ( f − 1 ( x ) , M \ f − 1 ( U ) ) ≥ 1 n .

(2) Let f be an ℐ -covering, π -mapping. Put d ′ ( x , y ) = d ( f − 1 ( x ) , f − 1 ( y ) ) for each x , y ∈ X . It is clear that d ′ is a d -function on X . Let a ∈ M , x ∈ X and n ∈ N . Suppose that B d ′ , n ( x ) = { y ∈ X : d ′ ( x , y ) < 1 n } .

(i) { B d ′ , n ( x ) } n ∈ N is a network at x in X for each x ∈ X .

Let U be an open neighborhood of x in X . Then, there exists n ∈ N such that d ( f − 1 ( x ) , M \ f − 1 ( U ) ) ≥ 1 n . If y ∉ U , then f − 1 ( y ) ⊆ M \ f − 1 ( U ) . Hence,

d ′ ( x , y ) = d ( f − 1 ( x ) , f − 1 ( y ) ) ≥ d ( f − 1 ( x ) , M \ f − 1 ( U ) ) ≥ 1 n .

So y ∉ B d ′ , n ( x ) . This implies that B d ′ , n ( x ) ⊆ U .

(ii) B d ′ , m ( x ) is an ℐ -sequential neighborhood of x for each x ∈ X and m ∈ N .

Let { x n } n ∈ N be a sequence in X with x n → ℐ x . Since f is an ℐ -covering mapping, there exists a sequence { a n } n ∈ N in M such that a n → ℐ a ∈ f − 1 ( x ) and each f ( a n ) = x n . Hence, there is I ∈ ℐ such that d ( a , a n ) < 1 m for each n ∈ N \ I . Thus, for each n ∈ N \ I ,

d ′ ( x , x n ) = d ( f − 1 ( x ) , f − 1 ( x n ) ) ≤ d ( a , a n ) < 1 m ,

i.e., the sequence { x n } n ∈ N is ℐ -eventually in B d ′ , m ( x ) . This means that B d ′ , m ( x ) is an ℐ -sequential neighborhood of x for each x ∈ X and m ∈ N .

By (i) and (ii), d ′ is an ℐ - s n -symmetric on X .□

The following lemma gives a very useful method to determine 1- ℐ -covering mappings, by which it can be seen that each open mapping on first-countable spaces is a 1- ℐ -covering mapping.

Lemma 4.9

[24, Lemma 4.3] Let ℐ be an ideal on N and f : X → Y be a mapping. If for each x ∈ X there is a decreasing network { B n } n ∈ N at x in X such that each f ( B n ) is an ℐ -sequential neighborhood of f ( x ) in Y , then f is a 1- ℐ -covering mapping.

The following is another main result in this article, which gives a positive answer to Problem 1.5 and shows that ℐ - s n -metrizable spaces can be characterized by certain images of semi-metric spaces.

Theorem 4.10

The following are equivalent for a regular space X:

X is an ℐ -sn-metrizable space.
X is a 1- ℐ -covering, π and σ -image of a semi-metric space.
X is an ℐ -covering, π and σ -image of an ℐ - s n -symmetric space.

Proof

We only need to prove that (1) ⇒ (2) and (3) ⇒ (1).

(1) ⇒ (2). Let X be an ℐ - s n -metrizable space. Then, X has an ℐ - s n -network P = ⋃ { P n : n ∈ N } , where each P n = { P α : α ∈ Λ n } is a locally finite family of subsets of X . For each n ∈ N , without loss of generality, we can suppose that P n is closed under finite intersections and X ∈ P n ⊆ P n + 1 . Each Λ n is endowed with the discrete topology. For the Tychonoff-product space ∏ n ∈ N Λ n , a point ( α n ) n ∈ N ∈ ∏ n ∈ N Λ n is abbreviated to ( α n ) . Put

M = { a = ( α n ) ∈ ∏ n ∈ N Λ n : { P α n } n ∈ N forms a network at some point x a in X } .

Defined f : M → X by f ( a ) = x a . Then, M is a metric space and f is continuous and surjective [11, Lemma 1.1].

(i) f is a 1- ℐ -covering mapping.

Let x ∈ X . Let X ∈ P x ⊆ P be an ℐ - s n -network at x in X . For each n ∈ N , since P n is point-finite and closed under finite intersections, there exists α n ∈ Λ n such that P α n = ⋂ { P ∈ P n : P ∈ P x } . Then, the set P α n is an ℐ -sequential neighborhood of x . Thus, it is obvious that the family { P α n } n ∈ N is a decreasing ℐ - s n -network at x in X . Put a = ( α n ) , then a ∈ f − 1 ( x ) . For each n ∈ N , let

B n ( a ) = { ( β i ) ∈ M : β i = α i for i ≤ n } .

Then, f ( B n ( a ) ) = P α n . In fact, if b = ( β i ) ∈ B n ( a ) , then f ( b ) ∈ ⋂ i ∈ N P β i ⊆ P β n = P α n . So f ( B n ( a ) ) ⊆ P α n . Conversely, let y ∈ P α n . Then, y ∈ P α i for each i < n , and there exists c ′ = ( γ i ′ ) ∈ M such that f ( c ′ ) = y . For each i ∈ N , put γ i = α i , if i ≤ n ; γ i = γ i − n ′ , if i > n . Then, { P γ i } i ∈ N is a network at y in X . Let c = ( γ i ) , then c ∈ B n ( a ) and f ( c ) = y , so y ∈ f ( B n ( a ) ) . Thus, P α n ⊆ f ( B n ( a ) ) . It is obvious that { B n ( a ) } n ∈ N is a decreasing neighborhood base at a in M . In view of Lemma 4.9, f is a 1- ℐ -covering mapping.

(ii) f is a σ -mapping.

Let ℬ = { B n ( a ) : a ∈ M and n ∈ N } . It is easy to see that ℬ a base of M and f ( ℬ ) = ⋃ n ∈ N { f ( B n ( a ) ) : a ∈ M } ⊆ ⋃ n ∈ N P n is σ -locally finite. This implies that f is a σ -mapping.

(iii) f is a π -mapping with respect to some semi-metric on M .

It follows from Corollary 4.7 that X is an ℐ - s n -symmetric space. Thus, f is a π -mapping with respect to some semi-metric on M from part (2) in Remark 4.3 and part (1) in Lemma 4.8.

(3) ⇒ (1). Let M be an ℐ - s n -symmetric space and f : M → X be an ℐ -covering, π and σ -mapping. It follows from Lemma 4.8 (2) that X is an ℐ - s n -symmetric space. Thus, X is an ℐ - s n f -countable space. In view of Theorem 3.3, it suffices to prove that X has a σ -locally finite ℐ - c s -network. Since f is a σ -mapping, there exists a base ℬ of M such that f ( ℬ ) = { f ( B ) : B ∈ ℬ } is a σ -locally finite family in X . So, we only need to prove that f ( ℬ ) is an ℐ - c s -network of X . Let S be a sequence ℐ -converging to x ∈ U with U open in X . Since f is an ℐ -covering mapping, there exists a sequence L in M ℐ -converging to some a ∈ f − 1 ( x ) and f ( L ) = S . Thus, there exists B ∈ ℬ such that a ∈ B ⊆ f − 1 ( U ) . So, L is ℐ -eventually in B , hence f ( L ) = S is ℐ -eventually in f ( B ) ⊆ U . This implies that f ( ℬ ) is a σ -locally finite ℐ - c s -network of X .□

The following example shows that semi-metrizability in Theorem 4.10 cannot be strengthened to metrizability. A mapping f : Y → X is called sequence-covering [27, Definition 2.1] if, whenever { x n } n ∈ N is a convergent sequence in X , there exists a convergent sequence { y n } n ∈ N in Y with each f ( y n ) = x n .

Example 4.11

There exists an ℐ - s n -metrizable space which is not an ℐ -covering, π -image of a metric space.

Proof

There exists a g -metrizable space X which is not a sequence-covering, π -image of a metric space [8, Example 3.1]. Put ℐ = ℐ fin . It follows from part (4) in Remark 2.5 that X is an ℐ - s n -metrizable space. By the definition of sequence-covering mappings, the sequence-covering mapping is ℐ -covering, thus X is not an ℐ -covering, π -image of a metric space.□

5 Conclusions

In this article, we introduce and study ℐ - s n -metrizable spaces, which is a generalization of s n -metrizable spaces and an application of ideal convergence in generalized metric theory. The following main conclusion is obtained, which gives several characterizations of ℐ - s n -metrizable spaces and establishes some mapping relations between ℐ - s n -metrizable spaces and semi-metric spaces.

Conclusion 5.1

Let ℐ be an admissible ideal on N . Then the following are equivalent for a regular space X .

X is an ℐ - s n -metrizable space.
X has a σ -discrete ℐ - s n -network.
X is an ℐ - s n f -countable and ℵ -space.
X is an FU- ℐ s n and s n -metrizable space.
X has a point-star ℐ - s n -network consisting of locally finite covers.
X is a 1- ℐ -covering, π and σ -image of a semi-metric space.
X is an ℐ -covering, π and σ -image of an ℐ - s n -symmetric space.

We will further discuss the following problems.

Problem 5.2

Let X be a regular space with a σ -hereditarily closure-preserving ℐ - s n -network. Is X ℐ - s n -metrizable?

Problem 5.3

Is every g -metrizable space an ℐ - s n f -countable space for each admissible ideal ℐ ?

The authors attempt to bring the concept of ideal convergence into the theory of generalized metric spaces. Given the profound background and widespread application of generalized metric theory, there are the following topics for researchers to discuss in the future:

the applications of these results in statistical convergence, summability theory, and probability theory;
the interesting properties of spaces defined by ideal convergence;
the mutual relationship between spaces and mappings in the sense of ideal convergence;
the topological algebraic structures in weakly topological groups with ideal continuity.

Acknowledgements

The authors would like to express their gratitude to the editors and the reviewers for their thoughtful comments and valuable suggestions.

Funding information: This research was supported by National Nature Science Foundation of China (No. 12171015), Nature Science Foundation of Fujian Province, China (No. 2023J011078), and Ningde Normal University (No. 2022FZ27, 2023T01).
Author contributions: The results of this article were jointly discussed and researched by all authors. The outline of the article was drafted by Shou Lin. Sections 1 and 5 were written by Shou Lin. Sections 2–4 were written by Xiangeng Zhou. All authors have reviewed and approved the paper.
Conflict of interest: The authors declare that they have no conflict of interests.
Data availability statement: Data sharing is not applicable to this article as no datasets were generated or analyzed during this study.

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Received: 2023-11-24

Revised: 2024-06-07

Accepted: 2024-07-29

Published Online: 2024-08-23

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

Articles in the same Issue

https://doi.org/10.1515/math-2024-0053

Keywords for this article

ideal convergence; ℐ-sn-networks; ℐ-cs-networks; ℐ-sn-metrizable spaces; semi-metric spaces; ℵ-spaces; ℐ-covering mappings; π-mappings; σ-mappings

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