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Review

Recent Progress on Point-Countable Covers and Sequence-Covering Mappings

by
Shou Lin
1,* and
Jing Zhang
2
1
Institute of Mathematics, Ningde Normal University, Ningde 352100, China
2
School of Mathematics and Statistics, Minnan Normal University, Zhangzhou 363000, China
*
Author to whom correspondence should be addressed.
Axioms 2024, 13(10), 728; https://doi.org/10.3390/axioms13100728
Submission received: 29 September 2024 / Revised: 16 October 2024 / Accepted: 17 October 2024 / Published: 21 October 2024
(This article belongs to the Special Issue Topics in General Topology and Applications)
Versions Notes

Abstract

:
This paper is dedicated to the memory of Professor Gary Gruenhage (1947–2023). This survey introduces the formation and early development of the topic of point-countable covers and sequence-covering mappings, and lists the recent progress of 38 questions on this topic proposed before 2015, which involve the theory of generalized metric spaces. These questions are related to point-countable covers and sequence-covering mappings, including point-countable covers with certain networks, sequence-covering mappings, images of metric spaces, and hereditarily closure-preserving families.

1. Introduction

This section briefly introduces the formation and early development of the topic of point-countable covers and sequence-covering mappings. All spaces in this paper are assumed to be T 2 , and all mappings are continuous and onto.
All the time, the problems in mathematics are a powerful driving force to lead the development of mathematics [1]. Since Open Problems in Topology [2,3] was published, general topology and related fields have seen huge development [4,5,6].
In 1961, P.S. Aleksandrov [7] proposed the idea of using mappings to study topological spaces, which involves using mappings as links to connect various types of spaces, and then, arranging research based on the differences between each type of space and mapping. In 1966, A.V. Arhangel’skiǐ [8] published a historical paper entitled “Mappings and Spaces”, which created a new era of classifying spaces by means of mappings. It provided specific problems for studying various spaces using mappings, known as the Aleksandrov–Arhangel’skiǐ problems [9]. The core content established the relationship between metric spaces and spaces with specific topological properties by mappings. These problems were outstanding contributions to general topology, and made the idea of classifying spaces by means of mappings become an important component of the theory of generalized metric spaces [10].
The essential relationship between constructing point-countable covers and mappings began in the 1960s.
Theorem 1
([11]). A topological space has a point-countable base if and only if it is an open s-image of a metric space.
The following metrization theorem shows the importance of point-countable covers.
Theorem 2
([12]). A Hausdorff countably compact space is metrizable if and only if it has a point-countable base.
A.V. Arhangel’skiǐ raised the following questions in 1966.
Question 1
([8]). What happens to metric spaces under closed mappings?
Question 2
([8]). How does one characterize, in intrinsic terms, quotient s-images of metric spaces?
Question 3
([8]). Is a point-countable base preserved by perfect mappings?
In 1966, N. Lašnev [13] gave the first intrinsic characterization of closed images of metric spaces, and the closed image of a metric space is now called a Lašnev space. In 1985, L. Foged obtained a better answer to Question 1 as follows.
Theorem 3
([14]). A regular space is a closed image of a metric space if and only if it is a Fréchet space with a σ-hereditarily closure-preserving k-network.
Quotient s-images of metric spaces were characterized by T. Hoshina in 1970 [15]. In 1984, G. Gruenhage, E.A. Michael, and Y. Tanaka [16] obtained another intrinsic characterization of quotient s-images of metric spaces. In 1987, Y. Tanaka obtained a better answer to Question 2 as follows.
Theorem 4
([17]). A Hausdorff space is a quotient s-image of a metric space if and only if it is a sequential space with a point-countable c s * -network.
In 1968, V.V. Filippov [18] proved that spaces with a point-countable base are preserved by perfect mappings, which gave an affirmative answer to Question 3. In 1972, E. Michael obtained a better answer to Question 3 as follows.
Theorem 5
([19]). A topological space has a point-countable base if and only if it is a countably bi-quotient s-image of a metric space.
In 1973, E.A. Michael and K. Nagami extended Theorem 1 to Theorem 6, and posed Question 4 ([2], Problem 394).
Theorem 6
([20]). A topological space has a point-countable base if and only if it is a compact-covering open s-image of a metric space.
Question 4
([20]). Is every quotient s-image of a metric space also a compact-covering quotient and s-image of a metric space?
In the 1970s and 1980s, point-countable covers had attracted a lot of attention from scholars in general topology. According to the current perspective, the following are representative works of generalized metric spaces determined by point-countable covers during this period:
(1)
D.K. Burke, E.A. Michael [21], On certain point-countable covers, 1976.
(2)
G. Gruenhage, E.A. Michael, and Y. Tanaka [16], Spaces determined by point-countable covers, 1984.
(3)
Y. Tanaka [17], Point-countable covers and k-networks, 1987.
In the process of classifying spaces by mappings, the mappings that have received the most attention are perfect mappings, closed mappings, compact-covering mappings, open mappings, pseudo-open mappings, and quotient mappings. And only a small amount of research has been conducted on other mappings. Analogous to the definition of compact-covering mappings, F. Siwiec introduced the definition of sequence-covering mappings [22]. Every open mapping defined on a first-countable space is sequence-covering [22]. The following theorem reflects the typical role of sequence-covering mappings.
Theorem 7
([22]). A topological space is a sequential space if and only if every sequence-covering mapping onto the space is quotient.
In addition, around Question 4, G. Gruenhage, E.A. Michael, and Y. Tanaka [16] defined another type of sequence-covering mapping and proved that the quotient s-image of a metric space must at least be a sequence-covering quotient s-image of a metric space (since the sequence-covering mapping in the sense of Siwice [22] is the sequence-covering mapping in the sense of Gruenhage, Michael, and Tanaka [16], the sequence-covering mapping in the sense of Gruenhage, Michael, and Tanaka is now called a pseudo-sequence-covering mapping [23]).
The main achievements of the theory of generalized metric spaces up to the 1980s were summarized in the following papers:
(1)
D.K. Burke, D.J. Lutzer [10], Recent advances in the theory of generalized metric spaces, Lecture Notes in Pure and Applied Mathematics, V. 24, 1976.
(2)
G. Gruenhage [24], Generalized metric spaces, Handbook of Set-theoretic Topology, 1984.
(3)
J. Nagata [25], Generalized metric spaces I, Topics in General Topology, 1989.
(4)
K. Tamano [26], Generalized metric spaces II, Topics in General Topology, 1989.
(5)
G. Gruenhage [27], Generalized metric spaces and metrization, Recent Progress in General Topology, 1992.
(6)
A.V. Arhangel’skiǐ [28], Paracompactness and metrization: the methods of covers in the classification of spaces, General Topology III, 1995.
(7)
R. Hodel [29], A history of generalized metrizable spaces, Handbook of the History of General Topology, V. 2, 1998.
R. Hodel also expressed the following optimistic views in A history of generalized metrizable spaces ([29], p. 570): “But more important perhaps is the fact that the study of generalized metrizable spaces is by no means complete; rather, is continues to grow with many new and important results appearing every year”. However, the undeniable fact is that by the late 1980s, research on this topic seems to have reached its extreme. In the 1990s, one of the key problems that plagued general topology was how to further develop the idea of classifying spaces by means of mappings.
The progress of the theory of generalized metric spaces in this century can be seen in the following surveys.
(1)
G. Gruenhage [30], Metrizable spaces and generalizations, Recent Progress in General Topology II, 2002.
(2)
G. Gruenhage [31], Generalized metrizable spaces, Recent Progress in General Topology III, 2014.
G. Gruenhage is a prominent contributor and extraordinary disseminator of the theory of generalized metric spaces. The special issue in Topology and its Applications in honor of his 70th birthday presented some highlights of Gary’s many contributions to general and set-theoretic topology, which is a testament to the impact his career has had on so many [32]. Undoubtedly, Generalized metric spaces [24] is one of the best references for general topology in the past 40 years, and his series of surveys [27,30,31] are the most authoritative overview of this topic in the past 60 years. It is his position at the pinnacle of this study direction and his immense influence on general topology that ensures the enduring vitality and widespread impact of this topic.

2. Spaces Determined by Networks and Mappings

It is precisely because of some questions in Open Problems in Topology [2] and Recent Progress in General Topology [5], and the mapping classes represented by sequence-covering mappings that new vitality has injected into the topic of generalized metric spaces in the critical period at the turn of the 20th and 21st centuries. These works were systematically summarized in Point-countable Covers and Sequence-sequence Mappings (the first edition, 2002) [33]. In the preface to the book [33], A.V. Arhangel’skiǐ stated:
“Though reading the monograph does not require much special background, the exposition in it goes far beyond the elementary level. It contains a rich collection of deep and beautiful results of highest professional level. The book not only brings the readers to the very first line of investigations in the theory of generalized metric spaces, but also contains many intriguing and important unsolved problems, some of them old and some new”.
“I would like to mention another joyful aspect of this monograph. Its appearance marks the success of a long period of development of general topology in China, it brings to the light important contributions to the mainstream of general topology made by a very creative group of Chinese mathematicians”.
This shows that the research on this topic not only highlights the value of the theory of spaces and mappings, but also provides an effective direction for the development of the theory of topological spaces in the early 21st century. From 2002 to 2015, 17 out of the 32 questions raised in the book had been solved.
The new version of the book Point-countable Covers and Sequence-covering Mappings (the second edition) [33] was published in 2015. It is devoted to the theory of spaces with point-countable covers and sequence-covering mappings on generalized metric spaces, including point-countable covers, sequences of point-finite covers, hereditarily closure-preserving covers, and star-countable covers. In the writing of the book and the research of related topics, a series of questions related to point-countable covers and sequence-covering mappings had arisen, some of which were later answered. To provide convenience for the readers who are interested to continue in-depth study of this topic, this article selects and analyzes 38 questions related to this topic, and uses them to illustrate the recent progress in this topic.
Let S = { 0 } ∪ { 1 / n : n ∈ N } , which is the usual convergent sequence. First, recall some basic definitions.
Definition 1.
Let X be a topological space, and P a family of subsets of X.
(1) 
P is called a k- n e t w o r k for X [34] if whenever K ⊂ V with K compact and V open in X, there exists a finite subfamily P ′ ⊂ P such that K ⊂ ⋃ P ′ ⊂ V .
(2) 
P is called a c s - n e t w o r k for X [35] if given a sequence { x n } n ∈ N converging to a point x ∈ X and a neighborhood V of x in X, then { x } ∪ { x n : n ≥ n 0 } ⊂ P ⊂ V for some n 0 ∈ N and some P ∈ P .
(3) 
P is called a c s * - n e t w o r k for X [36] if whenever { x n } n ∈ N is a sequence converging to a point x ∈ V with V open in X, then { x } ∪ { x n i : i ∈ N } ⊂ P ⊂ V for some subsequence { x n i } i ∈ N of { x n } n ∈ N and some P ∈ P .
(4) 
P is called a w c s * - n e t w o r k for X [37] if whenever { x n } n ∈ N is a sequence converging to a point x ∈ V with V open in X, then { x n i : i ∈ N } ⊂ P ⊂ V for some subsequence { x n i } i ∈ N of { x n } n ∈ N and some P ∈ P .
Definition 2.
Let P = ⋃ x ∈ X P x be a family of subsets of a topological space X satisfying the following: (a) for each x ∈ X , P x is a network of x in X [38], i.e., x ∈ ⋂ P x , and if x ∈ G and G is open in X, then P ⊂ G for some P ∈ P x ; (b) if U , V ∈ P x , then W ⊂ U ∩ V for some W ∈ P x .
(1) 
P is an s n - n e t w o r k for X [39] if each element P of P x is a sequential neighborhood of x for each x ∈ X , i.e., every sequence converging to the point x is eventually in P. The P x is called an s n -network at x in X.
(2) 
P is called a w e a k - b a s e for X [8] if, for every G ⊂ X , the set G is open in X whenever for each x ∈ G there exists a P ∈ P x such that P ⊂ G . The P x is called a w e a k - b a s e at x in X.
(3) 
A space X is s n f -(resp. g f -) c o u n t a b l e [8,39] if X has an s n -network (resp. a w e a k - b a s e ) P such that each P x is countable.
The following relations hold for a topological space [33]:
(1)
bases ⇒ weak-bases ⇒ s n -networks ⇒ c s -networks ⇒ c s * -networks ⇒ w c s * -networks.
(2)
bases ⇒ k -networks ⇒ w c s * -networks.
Definition 3.
Let X, Y be topological spaces, and f: X → Y a mapping.
(1) 
f is a c o m p a c t m a p p i n g (resp. s- m a p p i n g ) ([40], Definition 2.1.3) if each f − 1 ( y ) is compact (resp. separable) in X.
(2) 
f is a b o u n d a r y - c o m p a c t m a p p i n g (resp. a t m o s t b o u n d a r y - o n e m a p p i n g ) ([40], Definition 2.1.3) if each ∂ f − 1 ( y ) is compact (resp. at most one point) in X.
(3) 
f is a c o m p a c t - c o v e r i n g m a p p i n g [41] if each compact subset of Y is the image of some compact subset of X.
(4) 
f is a s e q u e n c e - c o v e r i n g m a p p i n g [22] if, whenever { y n } n ∈ N is a convergent sequence in Y, there exists a convergent sequence { x n } n ∈ N in X with each x n ∈ f − 1 ( y n ) .
(5) 
f is a p s e u d o - s e q u e n c e - c o v e r i n g m a p p i n g [16,23] if, whenever S is a convergent sequence containing its limit point in Y, there is a compact subset K in X such that f ( K ) = S .
(6) 
f is a s e q u e n t i a l l y q u o t i e n t m a p p i n g [42] if, whenever { y n } n ∈ N is a convergent sequence in Y, there exist a subsequence { y n i } i ∈ N of { y n } n ∈ N and a convergent sequence { x i } i ∈ N in X such that each x i ∈ f − 1 ( y n i ) .
(7) 
f is a 1- s e q u e n c e - c o v e r i n g m a p p i n g [39] if, for each y ∈ Y , there is an x ∈ f − 1 ( y ) such that whenever { y n } n ∈ N is a sequence converging to y in Y, there is a sequence { x n } n ∈ N converging to x in X with each x n ∈ f − 1 ( y n ) .
The following are obvious:
(1)
1-sequence-covering mappings ⇒ sequence-covering mappings ⇒ pseudo-sequence-covering mappings, and sequentially quotient mappings.
(2)
Boundary-compact closed mappings ⇒ compact-covering mappings ⇒ pseudo-sequence-covering mappings.
Definition 4.
Let X be a topological space.
(1) 
X is called a k-space [43] provided that a subset U ⊂ X is closed if and only if U ∩ K is closed in K for each compact subset K ⊂ X .
(2) 
X is called sequential ([40], Definition 1.6.15) provided that a subset U ⊂ X is closed in X if and only if a sequence in U converges to a point x in X, then x ∈ U .
(3) 
X is called Fréchet ([40], Definition 1.2.7) provided that a subset A ⊂ X and a point x ∈ A ¯ , then there is a sequence in A converging to x in X.
(4) 
X is called strongly Fréchet [22] provided that, for each decreasing sequence { A n } n ∈ N of subsets of X with a point x ∈ ⋂ n ∈ N A n ¯ , there is an x n ∈ A n for each n ∈ N such that the sequence { x n } n ∈ N converges to the point x.
The following are obvious [40]: First-countable spaces ⇒ strongly Fréchet spaces ⇒ Fréchet space ⇒ sequential spaces ⇒ k-spaces.
Readers may refer to General Topology [43] and Generalized Metric Spaces and Mappings [40] for notation and terminology not explicitly given here.

3. Point-Countable Covers

Let P be a family of subsets of a topological space X. P is called point-countable [40] if the family { P ∈ P : x ∈ P } is countable for each point x ∈ X .
Spaces with a point-countable base can be considered as the most beautiful ones with a point-countable cover. Many results on spaces with a point-countable cover can be traced back to certain properties of metric spaces, in particular, spaces having a point-countable base. Every space with a point-countable base is meta-Lindelöf, i.e., a space in which every open cover has a point-countable open refinement. The following results are related to this.
(1)
Every Fréchet space with a point-countable c s * -network is meta-Lindelöf [16,33].
(2)
Every regular Fréchet space with a point-countable k-network is meta-Lindelöf [16].
(3)
There exists a regular space with a point-countable weak-base, which is not meta-Lindelöf [16].
(4)
There exists a first-countable space with a σ -discrete k-network, which is not meta-Lindelöf ([33], Example 1.4.8).
Now, we introduce a concept as follows. Let X be a topological space and P a family of subsets of X. P is called a c s ′ -network ([33], Definition 2.1.15) for X if, whenever { x n } n ∈ N is a sequence in X converging to a point x ∈ U with U open in X, there exist an n ∈ N and a P ∈ P such that { x , x n } ⊂ P ⊂ U . Obviously, each c s * -network is a c s ′ -network in a topological space. However, there exists a regular space with a point-countable k-network that is not a c s ′ -network ([44], Remark 2.10).
Question 5
([33], Question 2.1.24). Is every Fréchet space with a point-countable  c s ′ -network meta-Lindelöf?
The answer to Question 5 is in the affirmative. The following result was obtained in [45] (Corollary 4.2): Every Fréchet space with a point-countable c s ′ -network is a hereditarily meta-Lindelöf space.
Every k-space with a point-countable k-network is preserved by closed mappings [37,46]. However, spaces with a point-countable k-network are not necessarily preserved by closed mappings [47]. As each compact subset of a space having a point-countable k-network is metrizable [16], we have the following question.
Question 6
([33], Question 2.3.7). Let X and Y be topological spaces, and f:  X → Y a compact-covering and closed mapping. Does the space Y have a point-countable k-network if X has a point-countable k-network and each compact subset of Y is metrizable?
Let P be a family of subsets of a topological space X. P is called c o m p a c t - c o u n t a b l e (resp., compact-finite) [40] if every compact subset of X meets at most countably (resp., finitely) many elements of P . It is known that every point-countable base of a topological space is compact-countable.
Question 7
([33], Question 2.3.16). Is every k-space with a compact-countable k-network preserved by closed mappings?
Michael and Nagami [20] posed the following famous problem ([2], Problem 394; see Question 4): Is every quotient s-image of a metric space also a compact-covering quotient and s-image of a metric space? However, Chen gave a negative answer to this problem, as shown in the following two examples:
(1)
There is a space which is a quotient compact image of a metric space, but not any compact-covering quotient and s-image of a metric space [48];
(2)
There is a regular space which is a quotient s-image of a metric space, but not any compact-covering quotient and s-image of a metric space under the condition with a σ ′ -set [49] (A set A in the real line is called a σ ′ -set [50] if, for every F σ -set F of the real line, there is an F σ -set H in the real line such that F ∩ H = ∅ and A ⊂ F ∪ H . An uncountable σ ′ -set can be constructed by assuming the continuum hypothesis [50]).
In order to characterize compact-covering s-images of metric spaces, we recall the concept of c f p -networks.
A family P of subsets of a topological space X is called a c f p -network for X [51] if, for every compact subset K of X and a neighborhood V of K in X, there exists a finite subfamily P ′ ⊂ P such that P ′ can be precisely refined by a finite cover of K consisting of closed subsets of K and ⋃ P ′ ⊂ V .
Obviously, every closed k-network is a c f p -network, and every c f p -network is a c s * -network and k-network in a topological space. However, there exist a regular space with a point-countable k-network that is not a c s * -network and a regular space with a point-countable c s * -network that is not a k-network ([33], Example 2.1.21).
Sequential spaces with a point-countable c s * -network can be characterized as quotient s-images of metric spaces [17], and k-spaces with a point-countable c f p -network can be characterized as compact-covering quotient and s-images of metric spaces [51,52]. Chen’s example shows that a sequential space with a point-countable c s * -network may not have any point-countable c f p -network.
Question 8
([33], Question 2.5.21).
(1) 
Does every sequential space with a compact-countable  c s * -network have a point-countable  c f p -network?
(2) 
Is there a regular space X which has a point-countable  c s * -network but no point-countable  c f p -networks under ZFC?
The answer to part (2) of Question 8 is in the negative. The following result was obtained in [53] (Proposition 2.4): There is a compact T 2 -space X with a point-countable c s -network such that X does not have any point-countable c f p -network.
A regular space is metrizable if and only if it has a σ -compact-finite base [54]. Closed images of metric spaces can be characterized by regular Fréchet spaces with a σ -compact-finite k-network [55]. It remains an open problem whether a regular space with a σ -compact-finite weak-base has a σ -locally finite weak-base [56]. It is known that a separable regular space with a σ -compact-finite k-network has a countable k-network under CH [57,58].
Question 9
([33], Question 4.1.24). Does every separable regular space with a  σ -compact-finite weak-base have a countable weak-base under ZFC?
A compact closed mapping is called a perfect mapping [43]. A countable-to-one perfect image of a space with a countable weak-base is not necessarily g f -countable ([33], Example 1.4.1).
Question 10
([33], Question 2.6.24). Is every space with a point-countable weak-base preserved by finite-to-one closed mappings?
Michael and Nagami [20] showed that each compact subset of a space with a point-countable base has a countable outer base. Let P = ⋃ x ∈ X P x be a weak-base for a space X in part (2) in Definition 2 and A ⊂ X . The family ⋃ x ∈ A P x is called an outer weak-base [59] of the set A in X.
Question 11
(([33], Question 2.7.20), [59]). Does every compact subset of a space with a point-countable weak-base have a countable outer weak-base?
The answer to Question 11 is in the affirmative. The following result was obtained in [53] (Proposition 2.12) and [60] (Lemma 2): Let X be a space with a point-countable weak-base. Then, every compact subset of X has a countable outer weak-base.
Let P be a cover of a topological space X. P is called point-regular [61] if x ∈ U and U is open in X, then { P ∈ P : x ∈ P ⊄ U } is finite. It is known that every point-regular c s * -network of a topological space is point-countable [23].
(1)
Spaces with a point-regular base can be characterized as (compact-covering) open and compact images of metric spaces [62].
(2)
Spaces with a point-regular weak-base can be characterized as sequence-covering quotient and compact images of metric spaces [23].
(3)
Spaces with a point-regular c s -network (resp. s n -network) can be characterized as sequence-covering compact images of metric spaces [63].
(4)
Spaces with a point-regular c s * -network can be characterized as pseudo-sequence-covering (and sequentially quotient) compact images of metric spaces [64].
Question 12
([33], Question 3.4.20). How to characterize spaces with a point-regular k-network by certain images of metric spaces?
Shakhmatov [65] and Watson [66] constructed independently the following famous example. There exists a completely regular pseudo-compact non-compact space X having a point-countable base. Uspenskii [67] proved that pseudo-compact spaces with a σ -point-finite base are metrizable.
Question 13
([33], Question 3.4.21). Is every pseudo-compact space with a point-regular weak-base metrizable?

4. Sequence-Covering Mappings

A finite-to-one closed mapping defined on a metric space is not necessarily a sequence-covering mapping [22]. Closed mappings on regular spaces in which each point is a G δ -set are sequentially quotient mappings ([33], Lemma 2.3.3).
Question 14
([33], Question 2.3.15). Is every closed mapping on spaces in which each point is a  G δ -set sequentially quotient?
The answer to Question 14 is in the negative. The following result was obtained in [53] (Proposition 2.2): There is a closed mapping f: X → S which is not sequentially quotient such that X is T 2 (non-regular) and every point of X is a G δ -set.
Siwice [22] showed that every open mapping on a first-countable space is a sequence-covering mapping. In fact, every almost-open mapping defined on a first-countable space is a 1-sequence-covering mapping ([33], Theorem 2.6.18). Let X and Y be topological spaces, and f: X → Y a mapping. f is called an almost-open mapping [68] if, for each y ∈ Y , there is an x ∈ f − 1 ( y ) such that whenever U is a neighborhood of x in X, then f ( U ) is a neighborhood of y in Y. There exists an open mapping on a Fréchet space such that it is not pseudo-sequence-covering ([33], Example 2.6.9), which gave a negative answer to ([22], part (a) in Problem 2.8).
Question 15
([33], Question 2.6.19). Is each almost-open mapping on a strongly Fréchet space a sequence-covering mapping?
The answer to Question 15 is in the negative. The following result was obtained in [53] (Theorem 2.6): There is an open mapping φ : X → S which is not sequence-covering such that X is countable and bi-sequential (A space X is said to be bi-sequential [19] if every ultrafilter A of X converging to a point x ∈ X contains a decreasing sequence { A n } n ∈ N converging to x. Every first-countable space is bi-sequential, and every bi-sequential space is strongly Fréchet [19]).
Sequence-covering mappings are 1-sequence-covering mappings under certain conditions. For example, boundary-compact sequence-covering mappings on first-countable spaces are 1-sequence-covering mappings ([33], Theorem 3.5.3), in particular, compact sequence-covering mappings on metric spaces are 1-sequence-covering mappings [69].
A regular space X is called g- m e t r i z a b l e  [70] if X has a σ -locally finite weak-base. A topological space is called submetrizable [24] if it can be mapped onto a metrizable topological space by a continuous one-to-one mapping.
Question 16
([33], Question 3.5.8). Let X and Y be topological spaces and f:  X → Y a boundary-compact sequence-covering mapping. Is f a 1-sequence-covering mapping if X satisfies one of the following conditions?
(1) 
Every compact subset of X has a countable  s n -network in X;
(2) 
Every compact subset of X has a countable outer  s n -network in X;
(3) 
X has a compact-countable  s n -network;
(4) 
X is a g-metrizable space;
(5) 
X is a submetrizable space.
The following relations hold in the conditions above: ( 4 ) ⇒ ( 3 ) ⇒ ( 2 ) . The answers to the space X satisfying the conditions (2)–(5) in Question 16 are in the negative. The following result was obtained in [53] (Proposition 2.14) and ([71], Example 3.5): There is a boundary-compact, sequence-covering mapping φ : X → Y which is not 1-sequence-covering such that X is a regular space with a countable weak-base. And the following result was obtained in [72] (Theorem 7): There exist a countable submetrizable space X and a compact sequence-covering map f: X → S ω such that f is not 1-sequence-covering. Where, S ω is called the sequential fan, i.e., it is the quotient space obtained by identifying all limit points of the topological sum of ω many convergent sequences.
Every closed sequence-covering mapping defined on a metric space is 1-sequence-covering ([33], Corollary 3.5.16). Furthermore, the following result was proved in [60] (Theorem 1): Let f: X → Y be a closed sequence-covering mapping. If X is a regular space having a point-countable weak-base, then f is 1-sequence-covering.
A topological space X is called developable [43] if there is a sequence { P n } n ∈ N of open covers of X such that the family { s t ( x , P n ) : n ∈ N } is a neighborhood base of x in X for each x ∈ X , where each st ( x , P n ) = ⋃ { P ∈ P n : x ∈ P } .
Question 17
([33], Question 3.5.20). Let X and Y be topological spaces and f:  X → Y a closed sequence-covering mapping. Is f a 1-sequence-covering mapping if X satisfies one of the following conditions?
(1) 
X is a developable space;
(2) 
Every compact subset of X has a countable neighborhood base in X;
(3) 
X is a submetrizable space.
The answer to part (3) of Question 17 is in the affirmative. The following result was obtained in [72] (Theorem 6): Let f: X → Y be a closed sequence-covering mapping, where X is a submetrizable space. Then, f is 1-sequence-covering.
Spaces with a point-countable base are preserved by countably bi-quotient s-mappings [73]. What kind of mappings can preserve spaces with a point-countable s n -network (resp. weak-base)? Spaces with a point-countable s n -network are preserved by countable-to-one 1-sequence-covering mappings [74]. Moreover, spaces with a point-countable s n -network can be characterized as 1-sequence-covering, at most boundary-one, and s-images of metric spaces ([33], Theorem 2.6.12).
Question 18
([33], Question 2.6.21), [74]). Is every space with a point-countable  s n -network preserved by 1-sequence-covering, at most boundary-one, and s-mappings?
The answer to Question 18 is in the negative. The following result was obtained in [53] (Theorem 2.8): There is a 1-sequence-covering, at most boundary-one and s-mapping φ : X → Y such that X has a σ -point-finite weak-base, but Y does not have any point-countable s n -network. The result also answers the following question in the negative (([33], Question 2.6.23), [75]): Is every space with a point-countable weak-base preserved by quotient, at most boundary-one, and s-mappings?
Sequence-covering and closed images of regular spaces with a point-countable weak-base are g f -countable [76].
Question 19
([33], Question 3.5.19). Let f:  X → Y be a sequence-covering and closed mapping. Is Y an  s n f -countable space if X has a point-countable sn-network?
Sequence-covering and closed images of regular spaces with a σ -compact-finite weak-base also have a σ -compact-finite weak-base [76].
Question 20
([33], Question 4.1.29), [77]). Does every sequence-covering and closed image of a space with a  σ -compact-finite  s n -network have a  σ -compact-finite sn-network?
The answers to Questions 19 and 20 are in the negative. The following result was obtained in [53] (Theorem 2.18): There is a sequence-covering closed mapping φ : Y → S ω such that Y has a σ -compact-finite s n -network. The sequential fan S ω is not s n f -countable ([53], Lemma 2.17).
Regular spaces with a point-countable base are preserved by sequence-covering and closed mappings [78].
Question 21
([33], Question 3.5.18). Let f:  X → Y be a sequence-covering and closed mapping. Does Y have a point-countable weak-base if X is a regular space with a point-countable weak-base?
We shall strengthen the forms of compact-covering and sequence-covering (resp. 1-sequence-covering) mappings. Let f: X → Y be a mapping.
(1)
f is called a 1- s c c - m a p p i n g  [59] if, for each compact subset K ⊂ Y , there is a compact subset L ⊂ X such that f ( L ) = K , and for each y ∈ K , there is a point x ∈ L such that if, whenever { y n } n ∈ N is a sequence converging to y, there exists a sequence { x n } n ∈ N converging to x in X with each x n ∈ f − 1 ( y n ) .
(2)
f is called an s c c - m a p p i n g  [59] if, for each compact subset K ⊂ Y , there is a compact subset L ⊂ X such that f ( L ) = K and if, whenever { y n } n ∈ N is a sequence converging to some point in K, there exists a sequence { x n } n ∈ N converging to some point in L with each x n ∈ f − 1 ( y n ) .
The following are known:
(1)
Every 1- s c c -mapping (resp. s c c -mapping) is a 1-sequence-covering (resp. sequence-covering) and compact-covering mapping.
(2)
Every compact-covering open mapping on a first-countable space is a 1- s c c -mapping [59].
(3)
Every s c c -mapping on a first-countable space is a 1- s c c -mapping ([33], Theorem 2.7.15).
(4)
There exists a 1-sequence-covering compact-covering mapping on a metric space which is not an s c c -mapping [59].
Question 22
(([33], Question 2.7.16), [59]). Is every s c c -mapping on compact spaces a 1- s c c -mapping?
The answer to Question 22 is in the negative. The following result was obtained in [53] (Theorem 2.10): Not every s c c -mapping of a compact space is 1- s c c .
Sequence-covering mappings are pseudo-sequence-covering mappings and sequentially quotient mappings. For a topological space X, every sequence-covering mapping onto the space X is a 1-sequence-covering mapping if and only if for each point x ∈ X there is a sequence { x n } n ∈ N converging to x in X such that the set { x n : n ∈ N } ∪ { x } is a sequential neighborhood of x in X [79]. On the other hand, every sequentially quotient mapping onto a space X is a 1-sequence-covering mapping if and only if there is a nontrivial convergent sequence in X [79].
Question 23
([33], Question 3.5.27). Give a characterization of a space X such that every pseudo-sequence-covering mapping onto the space X is a 1-sequence-covering mapping.
The following are some answers to Question 23. The following are equivalent for a space X ([53], Proposition 2.20):
(1)
Every pseudo-sequence-covering mapping onto X is 1-sequence-covering.
(2)
Every pseudo-sequence-covering mapping onto X is sequence-covering.
(3)
Every convergent sequence of X is a finite set.
(4)
Every mapping onto X is 1-sequence-covering.

5. Images of Metric Spaces

In the past fifty years, many interesting works have been induced by certain mappings on metric spaces. Obviously, every pseudo-sequence-covering mapping on a metric space is a sequentially quotient mapping ([33], Lemma 1.3.4). Every sequentially quotient and boundary-compact mapping on a developable space or a space with a point-countable base is a pseudo-sequence-covering mapping [80].
Question 24
([33], Question 3.5.21). Is every sequentially quotient s-mapping on a metric space a pseudo-sequence-covering mapping?
Every s n f -countable space can be characterized as a sequentially quotient and boundary-compact image of a metric space [74]. The sequentially quotient, at most boundary-one, and s-image of a metric space can be characterized as a space with a point-countable s n -network [74].
Question 25
(([33], Question 2.6.22), [74]). Let X be a sequentially quotient s-image of a metric space. Is X a sequentially quotient, boundary-compact, and s-image of a metric space if X is s n f -countable?
A topological space X is called feebly compact [81] if every locally finite family of open subsets of X is finite. It is clear that countably compact spaces are feebly compact, and that feebly compact spaces are pseudo-compact. Arhangel’skiǐ [82] showed that if a regular feebly compact space X is a pseudo-open s-image of a metric space, then X has a point-countable base. If a countably compact space X is a quotient s-image of a metric space, then X is metrizable [16]. Here, a mapping f: X → Y is called pseudo-open [83] if V is an open subset of X and f − 1 ( y ) ⊂ V for some y ∈ Y , then f ( V ) is a neighborhood of y in Y. Obviously, every pseudo-open mapping is a quotient mapping.
Question 26
([33], Question 2.2.12). Let X be a feebly compact space. Does X have a point-countable base if X is a quotient s-image of a metric space?
Liu [75] proved that a space is a quotient, at most boundary-one, and s-image of a metric space if and only if it is a quotient, at most boundary-one, and countable-to-one image of a metric space.
Question 27
(([33], Question 3.4.14), [74]). Is every quotient compact image of a metric space a countable-to-one quotient image of a metric space?
Although locally separable metric spaces are located between separable metric spaces and metric spaces, the images of locally separable metric spaces are very different from the images of separable metric spaces, or the images of metric spaces ([33], Example 5.1.23). It is easy to check that sequentially quotient s-images of metric spaces are equivalent to pseudo-sequence-covering s-images of metric spaces ([33], Theorem 2.2.8).
Question 28
([33], Question 5.1.8). Is every sequentially quotient s-image of a locally separable metric space equivalent to a pseudo-sequence-covering s-image of a locally separable metric space?
Although some intrinsic characterizations about quotient s-images of locally separable metric spaces have been obtained, most of these characterizations are complex and tedious ([33], Theorems 5.1.4, 5.1.7, and 5.1.9). It is still an open question: how to seek a nice intrinsic characterization of a quotient s-image of a locally separable metric space [84,85]. Some attempts have been made to solve the above question by the following questions.
Question 29
([33], Question 5.1.15). Is each closed Lindelöf subspace of a space X separable if the space X is a quotient s-image of a locally separable metric space?
Every first-countable subspace of a quotient s-image of a locally separable metric space is locally separable. However, there exists a space X which is a quotient s-image of a metric space such that every first-countable subspace of the space X is locally separable, but X is not any quotient s-image of a locally separable metric space [86].
Question 30
([33], Question 5.1.16). Let X be a quotient s-image of a metric space. If every first-countable subspace of X is locally separable and every closed Lindelöf subspace of X is separable, is X a quotient s-image of a locally separable metric space?
A topological space X is called a cosmic space (resp. an ℵ 0 -space) [41] if it has a countable network (resp. k-network). Cosmic spaces and ℵ 0 -spaces are very nice spaces represented as certain images of a separable metric spaces. Michael [41] proved that a space is a cosmic space if and only if it is an image of a separable metric space; and that a space is an ℵ 0 -space if and only if it is a compact-covering image of separable metric space.
It is clear that every quotient s-image of a locally separable metric space is a sequential space with a point-countable c s * -network consisting of cosmic subspaces.
Question 31
([33], Question 5.1.24). Is every sequential space with a point-countable c s * -network consisting of cosmic subspaces a quotient s-image of a locally separable metric space?
Let P be a cover of a topological space X. P is called a c s * -cover of the space X [87] if, for each convergent sequence S in X, there exists a P ∈ P such that some subsequence of S is frequently in P. P is called a k-cover of the space X [88] if every compact subset of X is covered by some finite subfamily of P . Spaces with a point-countable c s * -cover (resp. k-cover) consisting of ℵ 0 -subspaces are pseudo-sequence-covering (resp. compact-covering) s-images of locally separable metric spaces.
Question 32
(([33], Question 5.1.25), [89,90]).
(1) 
Is every pseudo-sequence-covering s-image of locally separable metric spaces a space with a point-countable c s * -cover consisting of ℵ 0 -subspaces? It is even unknown whether every quotient s-image of locally separable metric spaces has a point-countable c s * -network consisting of ℵ 0 -subspaces [84,91].
(2) 
Is every compact-covering s-image of locally separable metric spaces a space with a point-countable k-cover consisting of ℵ 0 -subspaces?
Sequentially quotient compact images of locally separable metric spaces are equivalent to pseudo-sequence-covering compact images of locally separable metric spaces [92]. A characterization of the images has been obtained, but it is quite complex [93]. Let X be a topological space and { P n } n ∈ N a sequence of covers of the space X. { P n } n ∈ N is called a point-star network [63] for X if the family { s t ( x , P n ) : n ∈ N } is a network of x in X for each x ∈ X , where each s t ( x , P n ) = ⋃ { P ∈ P n : x ∈ P } . It is known that a space is a pseudo-sequence-covering compact image of a metric space if and only if it has a point-star network of point-finite c s * -covers [64].
Question 33
([33], Question 5.2.4).
(1) 
Let X be a space with a point-star network of point-finite c s * -covers consisting of cosmic subspaces. Is X a pseudo-sequence-covering compact image of a locally separable metric space?
(2) 
Let X be a sequential space with a point-star network of point-finite c s * -covers consisting of cosmic subspaces. Is X a quotient compact image of a locally separable metric space?
Let ( X , d ) be a metric space and Y a topological space. A mapping f: X → Y is called a π-mapping [11] if d ( f − 1 ( y ) , X ∖ f − 1 ( U ) ) > 0 for each y ∈ Y and each neighborhood U of y in Y. Each compact mapping is a π -mapping on a metric space.
Question 34
(([33], Question 5.2.11), [94]). Is every quotient π -image of a locally separable metric space equivalent to a pseudo-sequence-covering quotient π -image of a locally separable metric space?
Quotient compact images of locally separable metric spaces are not necessarily quotient compact and compact-covering images of metric spaces [48]. Quotient π -images of separable metric spaces are pseudo-sequence-covering quotient and compact images of separable metric spaces [95]. Quotient compact and regular images of separable metric spaces are quotient compact and compact-covering images of separable metric spaces [69].
Question 35
([33], Question 3.3.24). Is every quotient compact image of a separable metric space a quotient compact and compact-covering image of a separable metric space?

6. Hereditarily Closure-Preserving Families

The research about hereditarily closure-preserving families mainly originates from closed images of locally finite families in a topological space. Let P be a family of subsets of a topological space X. P is called a hereditarily closure-preserving family [13] of the space X if, for each P ∈ P and each subset H ( P ) ⊂ P , the family { H ( P ) : P ∈ P } is closure-preserved, i.e.,
⋃ H ( P ) : P ∈ P ′ ¯ = ⋃ H ( P ) ¯ : P ∈ P ′
for each P ′ ⊂ P . P is called a point-discrete family [96] or a weakly hereditarily closure-preserving family [97] of the space X if, for each P ∈ P and each point p ( P ) ∈ P , the family { { p ( P ) } : P ∈ P } is closure-preserving, i.e., the set { p ( P ) : P ∈ P } is closed discrete in X. Obviously, every locally finite family is hereditarily closure-preserving, and every hereditarily closure-preserving family is point-discrete.
In regular spaces, the closure P ¯ = { P ¯ : P ∈ P } of a hereditarily closure-preserving family P is still hereditarily closure-preserving [98]. Generally, it is known that the closure of a hereditarily closure-preserving family is point-discrete [99].
Question 36
([33], Question 4.2.7). Is the closure of every hereditarily closure-preserving family still hereditarily closure-preserving?
The answer to Question 36 is in the negative. The following result was obtained in [53] (Proposition 2.22): There are a T 2 (non-regular) space X and a hereditarily closure-preserving family P in X such that P ¯ = { P ¯ : P ∈ P } is not hereditarily closure-preserving.
About the relations between a σ -point-discrete family and a σ -compact-finite family, it is known that a space with a σ -point-discrete s n -network (resp. k-network, w c s * -network, network) has a σ -compact-finite s n -network [77] (resp. k-network [100], w c s * -network [101], network [33]). However, there exists a space with a σ -point-discrete base (resp. weak-base) which does not have a σ -compact-finite base [97] (resp. weak-base [102]).
Question 37
([33], Question 4.1.19). Does every space with a σ -point-discrete c s -network have a σ -compact-finite c s -network or a σ -compact-finite c s * -network?
Sometimes, an s n f -countable space having certain c s -networks may imply that it has certain s n -networks. For example, a space with a point-countable s n -network if and only if it is an s n f -countable space with a point-countable c s -network [103].
Question 38
([33], Question 4.1.20). Does every s n f -countable space with a σ -point-discrete c s -network have a σ -point-discrete s n -network?
Because an s n f -countable space with a σ -point-discrete c s -network has a σ -compact-finite s n -network [77], a g f -countable space with a σ -point-discrete c s -network has a σ -compact-finite weak-base. However, there exists a g f -countable space with a σ -point-discrete w c s * -network which is not a space having a σ -point-discrete c s * -network ([33], Example 4.1.16(7)).
Question 39
(([33], Question 4.1.22), [101]). Does every g f -countable space with a σ -point-discrete c s * -network have a σ -compact-finite weak-base?
Liu and Ludwig [104] proved each closed mapping on a regular space with a σ -point-discrete base is a compact-covering mapping. Liu, Lin, and Ludwig [96] proved that every closed mapping on a regular space with a σ -point-discrete weak-base is also a compact-covering mapping under (CH).
Question 40
([33], Question 4.1.27). Is every closed mapping on a regular space with a σ -point-discrete s n -network a compact-covering mapping?
Question 41
(([33], Question 4.1.28), [77]). Is every space with a σ -point-discrete s n -network preserved by a sequence-covering closed mapping?
Lin, Liu, and Dai [91] proved that a regular space X with a σ -hereditarily closure-preserving k-network consisting of ℵ 0 -subspaces if and only if X has a σ -hereditarily closure-preserving k-network and each first-countable closed subspace of X is locally separable.
Question 42
([33], Question 4.3.11). Is each first-countable subset of a regular space with a σ -hereditarily closure-preserving k-network consisting of separable subspaces locally separable?

7. Conclusions

“Spaces and mappings” are an important research directions in general topology [28,105]. One of its frontier research topics is spaces with certain point-countable covers and sequence-covering mappings [33]. This article introduces the formation and development of this topic, and lists the background and recent progress of 38 questions proposed up to 2015. Among them, 10 questions have been solved and 3 questions have been partially solved.

Author Contributions

All authors participated in the research work of this topic. The research idea of this topic was first proposed by S.L. The answers to some questions were collected by J.Z. All authors have read and agreed to the published version of the manuscript.

Funding

This subject is supported by NNSF of China (No. 12171015, 12361012), NSF of Fujian Province of China (No. 2024J01804, 2024J01934) and the Research Project of Ningde Normal University, China (No. 2022FZ27).

Data Availability Statement

Data are contained within the article.

Acknowledgments

The authors would like to thank the editors and the reviewers for their thoughtful comments and constructive suggestions.

Conflicts of Interest

The authors declare no conflicts of interest.

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Lin, S.; Zhang, J. Recent Progress on Point-Countable Covers and Sequence-Covering Mappings. Axioms 2024, 13, 728. https://doi.org/10.3390/axioms13100728

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Lin S, Zhang J. Recent Progress on Point-Countable Covers and Sequence-Covering Mappings. Axioms. 2024; 13(10):728. https://doi.org/10.3390/axioms13100728

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Lin, Shou, and Jing Zhang. 2024. "Recent Progress on Point-Countable Covers and Sequence-Covering Mappings" Axioms 13, no. 10: 728. https://doi.org/10.3390/axioms13100728

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Lin, S., & Zhang, J. (2024). Recent Progress on Point-Countable Covers and Sequence-Covering Mappings. Axioms, 13(10), 728. https://doi.org/10.3390/axioms13100728

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