Researches on point-discrete families

Abstract

Based on the modern development of Metrization theorem for context, the main results obtained in recent ten years on point-discrete families are summarized. This paper mainly introduces the theory of the spaces with \sigma-point-discrete bases, the spaces with certain \sigma-point-discrete networks, and the relationship between the above spaces and the spaces with certain \sigma-compact-finite networks.

Graphical abstract

Keywords

Point-discrete families / compact-finite families / generalized metrizable spaces / k-net / weak bases

1 Introduction

In 1914, Hausdorff (DE, 1868–1942) published the influential work “Grundziige der Mengenlehre”(“Fundamentals of Set Theory”) [24], which developed the work of Hilbert (DE, 1862–1943), Weyl (DE, 1885–1955), Frechet (FR, 1878–1973), and Lliesz (Hungary, 1880–1956) and others to study geometric problems or abstract interspatial problems, giving a series of methods based on the neighborhood system. The definition of topological spaces based on the neighborhood system led to the study of some important topological spaces, among which Hausdorff spaces and Hausdorff metric are the most famous [10]. This marks the birth of general topology, and Hausdorff thus becomes the founder of general topology (see [4, 57]).

As an independent branch of mathematics, general topology has been fully developed and has contributed to the progress of mathematics and related disciplines. We dedicate this paper to the centenary of general topology, and to the memory of several national and international authors who have recently passed away: Mary Ellen Rudin (US, 1924–2013), Ernest A. Michael (US, 1925–2013), Jia-Lin Fang (1925–2014), and Guo-Jun Wang (1935–2013).

Metrizability and compactness have long been two central topics in general topology [1, 23]. In 1925, Urysohn (Russia, 1898–1924) succeeded in embedding regular spaces with countable bases into Hilbert squares, thus proving that the class of spaces is metrizable metric [55]. In 1950–1951, Bing (US, 1914–1986), Nagata (Japan, 1925–2007) and Smirnov (Russia, 1921–2007) independently gave sufficient conditions for the metrizability of topological spaces, respectively, and established the following well-known metrization theorem, now it is called the Bing-Nagata-Smirnov metrization theorem.

Theorem 1.1　 For a regular space X, the following conditions are equivalent:

(1) X is a quantifiable space;

(2) X has \sigma-discrete bases [5];

(3) X has \sigma \text{-}local\text{-}finite bases [48, 54].

The further development of the metricization theorem was mainly reflected in various generalizations to local finite set families and bases, which led to many important generalized classes of metric lift-off spaces, and gave a strong impetus to the flourishing of generalized metric sting space theory in the 1970s and 1980s [9, 16].

The family of compact-finite sets and the family of closed packet preserving sets in topological spaces are two natural generalizations of the family of locally finite sets. A subset of a space X is said to be compactly finite [6]. If every compact set in X intersects at most a finite number of sets in \mathcal{P}, \mathcal{P} is said to be closed-loop preserving of [45], if for any {\mathcal{P}}^{\mathrm{\prime }}\subset \mathcal{P}, there is \overline{\bigcup \{P:P\in {\mathcal{P}}^{\mathrm{\prime }}\}}=\bigcup \{\overline{P}:P\in {\mathcal{P}}^{\mathrm{\prime }}\}. It is easy to see that every locally finite family of sets is a compact finite family of sets that is closed under closure. In 1971, Boone [6] proved that regular spaces with \sigma \text{-}\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}\mathrm{a}\mathrm{c}\mathrm{t}\text{-}\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{e} bases are metrizable. However, as early as 1957, Michael [45] pointed out that the regular space with \sigma closed-loop-holding bases is not necessarily a metrizable space legacy. The subfamily P of the space X between the family of locally finite sets and the family of closure-preserving sets is said to be genetically maintained [25]. If for any {\mathcal{P}}^{\mathrm{\prime }}\subset \mathcal{P} and H(\mathcal{P})\subset P\in {\mathcal{P}}^{\prime }, there is \overline{\bigcup \{P:P\in {\mathcal{P}}^{\mathrm{\prime }}\}}=\bigcup \{\overline{H(P)}:P\in {\mathcal{P}}^{\mathrm{\prime }}\}, i.e., the family of sets \{H(P):P\in \mathcal{P}\}is closure-preserving. In 1975, Burke et al. [8] proved that a regular space with \sigma genetic closed-loop holding bases is metrizable. The space of regulars with \sigma genetic closed-loop retaining bases is metrically quantifiable. In 1986, Jiang [15] introduced the concept of a family of linear genetic closed-loop keepers and established a metric stinging theorem for the space of closed-loop keepers with trailing genes. The above series of results directly extended the Bing-Nagata-Smirnov Theorem. In [8], Burke et al. introduced and further investigated a generalization of the family of genetic closure preserving sets, which they called it weakly hereditarily closure-prese rving families. In 2008, at the suggestion of Arhangel'skii, Liu et al [43] renamed such families as point-discrete families. A family of subsets of a space X\in \mathcal{P} is said to be point-discrete [8, 43]. If for any x(P)\in P\in \mathcal{P},\{x(P):P\in \mathcal{P}\} is a closed discrete set. It is clear that the family of genetically closed preserving sets in T_1 space are point-discrete. Burke et al. [8] give counterexamples to show the existence of regular spaces with \sigma-point-discrete bases and proved that the regular k-spaces with \sigma-point-discrete bases are metrizable, which opened the way to study point-discrete sets—the first study on the family of discrete sets.

After Arhangel'skii [2] introduced the concept of network in 1959, some famous topologists have introduced the concept of special network (e.g., k-network). The concept of several special networks (k-net, weak bases, etc.) has defined many familiar classes of spaces, such as cosmic spaces, \sigma spaces, N-spaces, g-degree tractable spaces, etc., which greatly enriched and developed the theory of generalized degree tractable spaces. For a systematic discussion, please refer to [16–18, 36]. Although the study of spaces with various \sigma discrete (\sigmalocal-inite, \sigma genetic closure-preserving, \sigma closure-preserving) networks was completed in the mid-1990s, the study of spaces with various \sigma-point-discrete networks has been mostly carried out in the last decade. However, most of the research on spaces with various \sigma-point-discrete networks has been carried out only in the last decade, during which domestic scholars have invested a lot of efforts, constituting an important growth point in general topology and gaining. These research results are scattered in various journals and magazines, making it difficult for researchers who are new to the related fields to get a glimpse of them. The recent review on generalized metric spaces written by Gruenhage [20] is also scarce. For this reason, this paper categorizes and reviews the research results on the family of point-discrete sets and lists some unresolved problems. In this paper, we classify and review the research results on the family of point-discrete sets and list some unresolved problems for the reference of researchers who are interested in the related fields.

In this paper, all spaces are assumed to be at least Hausdorff's topological spaces and all mappings are continuous and full.

2 Space with \mathit{\sigma }-point-discrete basis

Firstly, we introduce two regular spaces with \sigma-point-discrete bases and non-metrizable quantization, which are the original motives for studying the family of point-discrete sets. The topological space X is called a sub-Lindelöf space if every open cover of X has a point countable open plus fine cover affine compact space and the space with point countable bases are sub-Lindelöf.

Example 2.1 [7]　There exist completely regular spaces with \sigma-point-discrete bases such that they are not sub-Lindelöf.

Example 2.2 [8]　There exists a genetic affine compact space with \sigma-point-discrete bases such that it is non-metrizable.

From the above two examples, it can be seen that the family of point-discrete sets is very different in nature from the family of genetic closure preserving sets: having \sigma-point-discrete. The regular space of scattered bases is not only not metrizable, but does not even satisfy the weak covering property of sub-Lindelöf; even if a strong covering property is attached. It is not enough to make the space with \sigma-point-discrete bases metrizable. The conditions under which a regular space with a \sigma-point-discrete basis can be quantified are not clear. Burke et al. [8] successively gave some sufficient conditions for the quantization of regular spaces with \sigma \text{-}\mathrm{p}\mathrm{o}\mathrm{i}\mathrm{n}\mathrm{t}\text{-}\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{c}\mathrm{r}\mathrm{e}\mathrm{t}\mathrm{e} bases.

A topological space X is called a k-space if for any A\subset X, A is a closed set in X if and only if for every compact set in X, K, K\cap A closed in K, X is called with countable compactness. For any x\in \overline{A}\subset X, there exists a countable set C\subset A such that x\in \overline{C}, X is called a q-space. For every x\in X, there exists a countable family of open neighborhoods \{U_n(x):n\in \omega \} of x such that a sequence x_n satiesfying x_n\in U_n(x) always has limit points. For x\in X, say \chi (x,X)=min\{|\mathcal{B}|:\mathcal{B}\mathrm{i}\mathrm{s}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{n}\mathrm{e}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{b}\mathrm{o}\mathrm{r}\mathrm{h}\mathrm{o}\mathrm{o}\mathrm{d}\mathrm{b}\mathrm{a}\mathrm{s}\mathrm{i}\mathrm{s}\mathrm{o}\mathrm{f}x\text{in}X\} is the identity of x in X, and \chi (X)=sup\{\chi (x,X):x\in X\} is the identity of X.

All first countable spaces, i.e., spaces with countable characteristics, are k-space, q-space and have countable compactness. However, the k-space property, q-space property, and countable compactness do not imply each other [46].

Theorem 2.1　 Let the regular space X have a \sigma-point-discrete basis, then the following conditions are equivalent:

(1) X is a metrizable space;

(2) X is a k-space [8];

(3) X has countable compactness [44];

(4) X is a q-space [44];

(5) (see [44]).

From the above theorem, it follows that the spaces constructed in Example 2.1 and Example 2.2 with \sigma-point-discrete basis are neither k-space or q-space, nor a countable compactness. Then it follows from (3) and (4) of Theorem 2.1 that a divisible or countably compact regular space with a \sigma-point-discrete basis is metrizable [44].

Problem 2.1 [43]　Is a regular space with a \sigma-point-discrete basis and satisfying the countable chain condition necessarily a metrizable space?

Problem 2.2 [43, 44]　Is the pseudo-compact space of a regular with a \sigma-point-discrete basis necessarily a metrizable space?

The property of a space with a \sigma-point-discrete basis is that the neighborhood assignment of a colorful space \left(X,\tau \right) is a function \varphi :X\to \tau, satisfying x\in \varphi (x)(\mathrm{\forall }x\in X). X is called a D-space [56]. The D-space is a hot space in general topology research in the last decade, and domestic scholars have made outstanding contributions [19].

Theorem 2.2 [44]　 Every regular space with \sigma-point-discrete basis is a D-space.

The mapping f:X\to Y is called a compact covering mapping if every compact subset of the space Y is a compact subset of the space X with respect to f mapping. A classical result on compact covering mappings is that a closed mapping on a degree lift space is a compact covering mapping [46].

Theorem 2.3 [44]　 Every closed mapping on a regular space with a \sigma-point-discrete basis is a compact covering mapping.

Each point of the metric space has a countable characteristic. Each point of the space with a \sigma-point-discrete basis is characterized by the following properties, which plays a key role in characterizing the product property.

Theorem 2.4 [7]　If the space X has a \sigma-point-discrete basis, then for each non-isolated point x,\chi (x,X) has a countable cotails.

Theorem 2.5 [7]　 Let the spaces X and Y have \sigma-point-discrete bases, then X\times Y has \sigma-point-discrete bases if and only if \chi (x,X)=\chi (y,Y) for non-isolated point x and y in X and Y.

However, the product and mapping properties of the space with \sigma-point-discrete basis are inferior to those of the space with \sigma \text{-}\mathrm{l}\mathrm{o}\mathrm{c}\mathrm{a}\mathrm{l}\text{-}\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{e} basis.

Example 2.3 [7]　There exist regular spaces X and Y with \sigma-point-discrete bases such that X\times Y does not have a \sigma \text{-}\mathrm{p}\mathrm{o}\mathrm{i}\mathrm{n}\mathrm{t}\text{-}\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{c}\mathrm{r}\mathrm{e}\mathrm{t}\mathrm{e} basis.

Example 2.4 [43]　There exists any finite product of regular spaces X, Y with \sigma-point-discrete basis. However, X^{\omega } does not have a \sigma-point-discrete basis.

Example 2.5 [7]　A complete mapping does not preserve a space with a \sigma-point-discrete basis.

A topological space X is said to have the point G_{\delta } property or to have countable pseudo equivalence if every single point set in X is a G_{\delta } set of X.

Jiang [15] proved that a regular space with \sigma-linear genetic closed-loop preserving bases is metrizable if and only if it has countable pseudo-features. The following problem is the most interesting problem in this section, and if it is answered in the affirmative, then the answer to Problem 2.2 is also affirmative.

Problem 2.3 [43, 44]　Does a regular space with a \sigma-point-discrete basis necessarily have the point G_{\delta } property?

3 k-net and related networks

Networks and k-net are the central concepts of generalized metric space theory [16, 47]. In the 1970s–1990s, a group of general topologists replaced the compact set in the concept of k-net with a proper convergence sequence and introduced the concepts of cs-net, cs*-net, and wcs*-net. Nowadays, these networks have been proved to play an important role in the theory of generalized metric spaces, especially in the study of various images and topological algebra of the inscribed metric spaces algebra [27, 35] on networks with \sigma-local finite \sigma-closed-hold or \sigma-generic closed-hold. A systematic study of space classes of networks (k-net, cs-net, wcs-net) can be found in [16, 35, 36]. In the following, we focus on the progress of the study of spatial classes with these \sigma-point-discrete networks since 2000.

Definition 3.1　Let \mathcal{P} be a cover of a topological space X.

(1) \mathcal{P} is called a network or simply a net of X [2]. For any open set U and x\in U in X, there exists P\in \mathcal{P}, such that x\in P\subset U.

(2) \mathcal{P} is called a k-network of X [50]. For any compact set K and open set U in X satisfying K\subset U, there exists a finite subfamily {\mathcal{P}}^{\mathrm{\prime }}\subset \mathcal{P} such that K\subset \bigcup {\mathcal{P}}^{\mathrm{\prime }}\subset U holds.

(3) \mathcal{P} is called a cs-net of X [21]. For any open set U, x\in U, and any sequence L converging to x, there exists P\in \mathcal{P} such that x\in P\subset U and L ends in P.

(4) \mathcal{P} is called a cs*-net of X [12]. For any open set U, x\in U, and any sequence L that converges to x, there exists a subcolumn L′ of L and P\in \mathcal{P} such that x\in P\subset U and L^{\mathrm{\prime }}\subset P\subset U.

(5) \mathcal{P} is called a wcs*-net of X [37]. For any open set U, x\in U, and any sequence L that converges to x, there exists a subcolumn L′ of L and P\in \mathcal{P} such that L^{\mathrm{\prime }}\subset P\subset U.

It is easy to see that (1) base \Rightarrow cs-net \Rightarrow cs*-net \Rightarrow wcs*-net \Rightarrow net; (2) base \Rightarrow k-net \Rightarrow wcs*-net.

The following Lemma is the basic method to deal with the family of known discrete sets, while establishing the basic relationship between the family of point-discrete sets and the family of compact-finite sets.

Theorem 3.1 [38, 49]　 Let \mathcal{P} be a point-discrete family in space X.

(1) If K is a compact set of X, then there exists a finite subset F of K such that\{P\in \mathcal{P}:P\cap (K\setminus F)\ne \mathrm{\varnothing }\} is finite.

(2) Let D=\{x\in X:\mathcal{P}\text{in}x\text{is not point-finite}\}. Then \{P\setminus D:P\in \mathcal{P}\}\cup \{\{x\}:x\in D\} is compact-finite.

The above (2) shows that it is easy to verify that the family of point finite and point-discrete sets is the family of compact-finite sets, and the family of compact-finite sets in k-space is the family of point-discrete sets. Using the above lemma and the construction method in the proof of the main result of [38], the following corollary holds.

Corollary 3.1　(1) A space with a \sigma-point-discrete net has a \sigma-compact-finite net, so each compact subset of such a space is metrically liftable of such a space is metrizable.

(2) A space with a \sigma \text{-}point\text{-}discrete k-net has a \sigma \text{-}compact\text{-}finite k-net [51].

(3) The space with \sigma-point-discrete wcs*-net is equivalent to the space with \sigma \text{-}point\text{-}discrete k-net [52].

(4) The space with \sigma-compact-finite wcs*-net is equivalent to the space with \sigma \text{-}compact\text{-}finite k-net.

For completeness, the following is a brief explanation of why (1) and (4) hold. Let \mathcal{P}=\bigcup _{n\in \mathbb{N}}{\mathcal{P}}_n be a \sigma-point-discrete net of the space X network, where each pill is a family of point-discrete sets of X. For each n\in \mathbb{N}, let D_n=\{x\in X:{\mathcal{P}}_n\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}\mathrm{d}\mathrm{u}\mathrm{n}\mathrm{d}\mathrm{a}\mathrm{n}\mathrm{c}\mathrm{y}\mathrm{i}\mathrm{n}x\mathrm{i}\mathrm{s}\mathrm{n}\mathrm{o}\mathrm{t}\mathrm{p}\mathrm{o}\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{e}\}, {\mathcal{F}}_n=\{P\setminus D_n:P\in {\mathcal{P}}_n\}\cup \{\{x\}:x\in D_n\}. Denote \mathcal{F}=\bigcup _{n\in \mathbb{N}}{\mathcal{F}}_n. By Lemma 3.1, \mathcal{F} is \sigma-compact-finite.

For any open set U of X and the point x\in U, there exists n\in \mathbb{N}\text{and}P\in {\mathcal{P}}_n\text{such that}x\in P\subset U. If x\in D_n, take F=x; if x\notin D_n, take F=P\setminus D_n. Then F\in {\mathcal{F}}_n\subset \mathcal{F}\text{and}x\in F\subset U. Therefore, it follows that \mathcal{F} is the net of X, i.e., X has \sigma-compact-finite nets. The restricted net is metrizable since the compact space with countable nets is metrizable, so each compact subset of X is metrizable, so (1) holds.

Since in every space where a compact subset is metrizable, the wcs*-net with countable points is a k-net (see [35, Lemma 2.1.6]), so the space with \sigma-compact-finite nets also has \sigma \text{-}\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}\mathrm{a}\mathrm{c}\mathrm{t}\text{-}\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{e} k-net. Since then, (4) holds.

It follows that the \sigma-point-discrete property is stronger than \sigma-compact-finite properties for spaces considered with specific properties of nets, k-net or wcs*-net.

The fan space s_{{\omega }_1} is also the quotient obtained by gluing the topology of {\omega }_1 a nontrivial convergent sequence and all convergent points in the space into a single point. The space similarly to the sequence fan s_{\omega } is the quotient space obtained by gluing the topology of \omega nontrivial convergent sequences and all convergent points in the space into a single point. The preliminary properties of the two spaces mentioned above can be found in [36, Example 1.8.7].

Example 3.1　Fan space S_{{\omega }_1}: with a \sigma-point-discrete cs*-net but neither a \sigma \text{-}\mathrm{p}\mathrm{o}\mathrm{i}\mathrm{n}\mathrm{t}\text{-}\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{c}\mathrm{r}\mathrm{e}\mathrm{t}\mathrm{e} cs-net nor a \sigma \text{-}\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}\mathrm{a}\mathrm{c}\mathrm{t}\text{-}\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{e} cs*-net.

The following interproblem is the core interproblem of this section.

Problem 3.1 [52]　Does a space with \sigma-point-discrete cs-net definitely have a \sigma \text{-}\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}\mathrm{a}\mathrm{c}\mathrm{t}\text{-}\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{e} cs*-net?

We do not even know if the space with \sigma-point-discrete cs-net necessarily has \sigma-compact-finite cs*-net. The related problem is whether the space with \sigma-compact-finite cs*-net has \sigma-compact-finite cs-net. Example 5.1 later shows that the space with \sigma-compact-finite nets (k-net, cs-net, cs*-net, wcs*-net) correspondingly cannot be introduced into the space with \sigma-point-discrete nets (k-net, cs-net, cs*-net, wcs*-net) of the space.

Definition 3.2 [11]　A subset P of a space X is said to be a sequential neighborhood of point-work x\in X if any sequence in X that converges to x eventually ends up in P. A subset U\in X is called a sequence open set if U is a sequence neighborhood of each of these points. X is called the sequence space if every sequence open set in X is open.

Let X be a space X in which all sequence open sets form a new topology on X. The space obtained by giving X this topology is denoted as \sigma X, and is called the sequence reflection topology of X. It is easy to verify that the space X is a sequence space if and only if X=\sigma X.

Theorem 3.1 [26]　 Let the space X have \sigma-point-discrete pair nets. If \sigma X does not contain closed subspaces homozygous s_{\omega }, then X has \sigma \text{-}compact\text{-}finite cs*-net.

For a space X, if \sigma X does not contain a closed subspace homozygote in s_{\omega }, then X does not contain a closed subspace homozygote in s_{\omega }. The following problem is related to Problem 3.1.

Problem 3.2 [26]　Let the space X does not contain a closed subspace homomorphism to S_{\omega }. If X has a \sigma-point-discrete cs*-net (or cs-net). Does X have a \sigma \text{-}\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}\mathrm{a}\mathrm{c}\mathrm{t}\text{-}\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{e} cs*-net (or cs-net)?

Theorem 3.2 [26]　 Let X be a regular space with a \sigma-point-discrete wcs*-net if X^{\omega } is a k space. Then X is a metrizable space.

The above theorem is correct for regular spaces with \sigma-point-discrete bases, which is a result in the paper [43]. However, it is no longer holds for regular spaces with \sigma-point-discrete nets. For example, take X to be any first countable unmeasurable space with a countable net (see [36, Example 1.8.3]), then X^{\omega } is the first countable space, and thus X^{\omega } is a k-space.

Recall that several mappings. Set the mapping f:X\to Y. f is called a sequence covering mapping if every convergent sequence in Y is a sequence of convergent sequences in X, f is called the sequence quotient mapping. For every convergent sequence S in Y, there exists a convergent sequence in X sequence L such that f(L) is a subsequence of S. In terms of mapping holds, it is easy to verify that since closed mappings hold a family of discrete sets of points: the closed mapping of the sequence cover holds the space with a \sigma-point-discrete cs-net; the closed mapping of the sequence quotient holds the space with a \sigma-point-discrete cs*-net (wcs*-net, k-net); the closed mapping of sequence quotient holds the space with \sigma-points of discrete nets.

4 Weak bases and their promotion

As a generalization of basis, the concept of weak basis was introduced by Arhangel'skii in his classical report “Mapping and Space” in 1966 [3]. The class of spaces with various weak bases is an important object of study in the theory of generalized metric spaces with \sigma-local finite (\sigma genetic closure-preserving, \sigma closure-preserving) space classes with weak bases are studied in [35, 36]. An important milestone of weak bases result is that in 2005, Liu [41] proved that regular spaces with \sigma genetic closure-preserving weak bases have \sigma-local-finite weak bases. This section focuses on the important research results in the last decade on the class of spaces with \sigma-point-discrete weak bases and related spaces.

Definition 4.1　Let \mathcal{B}=\bigcup \{{\mathcal{B}}_x(n):x\in X,n\in \omega \} be a family of sets in a space X, satisfying that for any x\in X,n\in \omega ,{\mathcal{B}}_x(n) is closed under finite intersection and x\in \bigcap {\mathcal{B}}_x(n).

(1) If for every subset A of X, A is an open set if and only if for any x\in A and n\in \omega, there exists {\mathcal{B}}_x(n) such that B\subset A holds. Then \mathcal{B} is said is said to be a weak base {\mathrm{\aleph }}_0 of X [42]. Then, if each {\mathcal{B}}_x(n) is countable, then the space X is weakly countable [53].

(2) In the definition of weak basis {\mathrm{\aleph }}_0, if the definition of {\mathcal{B}}_x(n)={\mathcal{B}}_x(1) holds for every n\in \omega, then \mathcal{B} is said to be a weak basis of X [3]. In this case, the space X is said to be a countable weakly basis if each of the names {\mathcal{B}}_x(1) is countable [3].

Definition 4.2　Let \mathcal{P}=\bigcup \{{\mathcal{P}}_x:x\in X\} be a family of sets in the space X, satisfying the following conditions. For any x\in X, {\mathcal{P}}_x is closed under finite intersection. If every element in {\mathcal{P}}_x is a sequential neighborhood of x and for any open neighborhood U of x, there exists P\in {\mathcal{P}}_x such that x\in \mathcal{P}\subset U holds. Then \mathcal{P} is said to be a sn-net of X [33]. At this point, if every {\mathcal{P}}_x is countable, then the space X is said to be the first countable [15, 34].

It is easy to see that: (1) Base-weak basis \Rightarrow {\mathrm{\aleph }}_0 weak basis \Rightarrow cs*-net;

(2) Weak base \Rightarrow sn-nets; and in the sequence space, sn-net is now a weak base;

(3) First countable space \Rightarrow weak first countable space\Rightarrow weakly proposed first countable space \Rightarrow sequence space \Rightarrow k-space; and weakly first countable space \iff sn the first countable sequence space.

The following result is a nontrivial theorem on the family of \sigma-point-discrete sets whose inverse proposition does not hold, see Example 5.1.

Theorem 4.1 [31]　 Spaces with \sigma-point-discrete sn-net have \sigma-compact-finite sn-net.

From this, a further exact connection with the space of discrete cs-net with \sigma-points can be obtained. Note that if X is an sn first countable space, then \sigma X does not contain the closed subspace homozygous S_{\omega }, see [35].

Theorem 4.2　 The following conditions are equivalent:

(1) X has a \sigma \text{-}point\text{-}discrete sn-net;

(2) X is a sn first countable space with a \sigma \text{-}point\text{-}discrete cs-net [30];

(3) X has a \sigma-point-discrete cs-network and \sigma X without closed subspaces is homogeneous with S_{\omega }, see [28].

Example 4.1　There exists a regular space X with countable cs-net such that X does not contain closed subspaces homogeneous to S_{\omega }. However, \sigma X is homogeneous to S_{\omega }, see [34, Example 3.19]. This shows that \sigma X in condition (3) of Theorem 4.2 is not reducible to X.

Corollary 4.1　 The following conditions are equivalent:

(1) X has a \sigma \text{-}compact\text{-}finite weak base.

(2) X is a weak first countable space with \sigma \text{-}point\text{-}discrete weak bases [38].

(3) X is a k space with \sigma \text{-}point\text{-}discrete sn-net [30].

(4) X is a k space with \sigma \text{-}compact\text{-}finite sn-net.

(5) X is a weakly first countable space with \sigma \text{-}point\text{-}discrete cs-net [43].

If X is regular, then the above conditions are also equivalent to the following conditions under the continuum assumption.

(6) X is a space with \sigma-point-discrete weak basis and countable compactness [43].

The equivalence of the above condition (4), which is not found in the literature, is stated as follows. By Theorem 4.1, we know (3) \Rightarrow (4). If X is a k-space with \sigma-compact k-space with finite sn-nets, as X is a metrizable k-space for each compact subset, so X is a sequence space. Thus, the sn-net is a weak basis of X such that X has a \sigma \text{-}\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}\mathrm{a}\mathrm{c}\mathrm{t}\text{-}\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{e} weak basis, i.e., (4) \Rightarrow (1).

It is not known whether the cs-net in Theorem 4.2 (2) and Corollary 4.1, (5) can be reduced to a cs*-net, so there is the following problem.

Problem 4.1　Does a weak first countable (sn first countable) space with \sigma-point-discrete pairwise nets have \sigma \text{-}\mathrm{p}\mathrm{o}\mathrm{i}\mathrm{n}\mathrm{t}\text{-}\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{c}\mathrm{r}\mathrm{e}\mathrm{t}\mathrm{e} weak basis (sn-nets)?

In contrast to Theorem 2.2, we have the following problem.

Problem 4.2　Is every regular space with a \sigma-point-discrete weak base a D-space?

Theorem 4.3　 Let f:X\to Y be a closed mapping covered by a sequence. If X is a regular space with \sigma-compact-finite weak bases, then Y has \sigma-compact-finite weak bases.

The following examples deepen our understanding of the properties of mappings of spaces determined by weak bases and sn-nets.

Example 4.2　(1) A perfect mapping does not preserve the space with \sigma-point-discrete weak bases, which is only necessary to verify the properties of the mapping does not have a \sigma-point-discrete weak basis in Example 3.3, given by Burke and Davis [7].

(2) There exists a perfect mapping f:S_2\to S_{\omega }, where S_2 (i.e., the Arens space) has a countable weak basis and S_{\omega } is not the first countable space of sn countable space, see [35, Example 1.5.1 and Example 1.5.2].

(3) Closed mappings on spaces with \sigma-compact-finite sn-nets are not necessarily compact covering mappings, see [35, Example 2.2.2].

The following mapping problem is also interesting.

Problem 4.3 [30]　Does a closed mapping of sequence coverings preserve the space with \sigma \text{-}\mathrm{p}\mathrm{o}\mathrm{i}\mathrm{n}\mathrm{t}\text{-}\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{c}\mathrm{r}\mathrm{e}\mathrm{t}\mathrm{e} sn-nets or with \sigma \text{-}\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}\mathrm{a}\mathrm{c}\mathrm{t}\text{-}\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{e} sn-nets?

Noting that the image space in Example 4.2 (1) has a \sigma-point-discrete weak basis, we have the following problem.

Problem 4.4　Does a perfect map preserve the space with \sigma-point-discrete weak bases?

The following problem is related to Theorem 2.3.

Problem 4.5　Is a closed mapping on a regular space with a \sigma-point-discrete sn-net a compact covering mapping?

The following conclusion is similar to Corollary 4.1 for the weak bases of {\mathrm{\aleph }}_0.

Theorem 4.4 [52]　 The following conditions are equivalent:

(1) X has a \sigma \text{-}compact\text{-}finite weak basis;

(2) X is a k-space with \sigma \text{-}point\text{-}discrete weak bases;

(3) X is a weakly fitted first countable space with a \sigma \text{-}point\text{-}discrete weak basis.

In addition, under the continuum assumption, the following further result, which is related to Problem 4.5, partially weakens the conditions of Theorem 2.3.

Theorem 4.5 [52] (CH)　 Every closed mapping on a regular space with \sigma-point-discrete weak bases is a compact covering mapping.

The most expected solution in this paper is the following problem.

Problem 4.6 [39]　Does a regular space with \sigma-compact-finite weak bases have locally \sigma finite weak bases?

The following two interproblems are related to Problem 2.3 and Problem 2.1, respectively.

Problem 4.7 [30, 43]　Does a regular space with \sigma-point-discrete weak bases (sn-nets) have the point G_{\delta } property?

Problem 4.8 [43]　Is a pseudo-compact space with \sigma-point-discrete weak bases metrizable?

This section concludes with a discussion of the partial role of countability for families of point-discrete sets.

After Theorem 2.1, it has been shown that the following problem answers in the affirmative for bases. It has been shown in [40, 52] that under the continuum hypothesis, the answer to the question remains positive. Even if we strengthen “point dispersion” to “compact-finiteness”, we still cannot give a definite answer to the following question in general.

Problem 4.9 [40, 52]　Does a divisible regular space with \sigma-points of discrete weak bases ({\mathrm{\aleph }}_0 weak bases) have countable weak bases ({\mathrm{\aleph }}_0 weak bases)?

The following example shows that the answer to the question is negative if the weak or weak group in Problem 4.9 is replaced by an sn-net.

Example 4.3　There exist separable regular spaces with \sigma-local-finite sn-nets that do not have countable sn-nets, see [31, Example l]. The topological space X is called {\mathrm{\aleph }}_1-compact if every set in X with base {\mathrm{\aleph }}_1 has a cluster. Lindelöf spaces and genetically separable partition spaces are both {\mathrm{\aleph }}_1-compact spaces.

Theorem 4.6　 Let X be an {\mathrm{\aleph }}_1-compact space.

(1) If X has a \sigma \text{-}point\text{-}discrete net, then X has a countable net [44].

(2) If X has a \sigma \text{-}point\text{-}discrete k-net, then X has a countable k-net [32].

(3) If X has a \sigma \text{-}point\text{-}discrete sn-net, then X has a countable sn-net [13].

(4) If X has a \sigma-point-discrete {\mathrm{\aleph }}_0 weak basis then X has a countable {\mathrm{\aleph }}_0 weak basis [52].

(5) If X has a \sigma-point-discrete weak basis then X has a countable weak basis [38].

(6) If X has a \sigma \text{-}point\text{-}discrete basis, then X has a countable basis [44].

The above theorem formally does not involve the cs-net, cs*-net and wcs*-net which are of interest in this paper, but they are embedded in condition (2), as the following properties of the space are equivalent to each other: countable cs-net countable cs*-net, countable wcs*-net and countable k-net. In fact, if \mathcal{P} is a countable wcs*-net of space X, it is easy to verify set \{\bigcup \mathcal{F}:\text{finite}\mathcal{F}\subset \mathcal{P}\} is both a countable cs-net and k-net of X.

Problem 4.10　Let X be a divisible regular k-space if X has a \sigma-point-discrete k-net, then does X have a countable k-net?

5 Summary

To conclude, we use a diagram to summarize the basic relationships between the main spaces covered in this paper. In order to avoid excessive line crossings, this diagram does not list the corresponding contents of the spacious and weak bases, and it is easy for the reader to make additions from the basic relations introduced in the paper.

Some relevant examples not described in the above sections are added below.

Example 5.1　Fortissimo space (see [52, Example 2.11]): regular space with \sigma \text{-}\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}\mathrm{a}\mathrm{c}\mathrm{t}\text{-}\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{e} sn-nets, but without \sigma \text{-}\mathrm{p}\mathrm{o}\mathrm{i}\mathrm{n}\mathrm{t}\text{-}\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{c}\mathrm{r}\mathrm{e}\mathrm{t}\mathrm{e} nets.

Example 5.2　Arens space S_2 (see [36, Example 1.8.61]): regular k-spaces with countable weak bases, but not metrizable. By Theorem 2.1, S_2 does not yet have a \sigma \text{-}\mathrm{p}\mathrm{o}\mathrm{i}\mathrm{n}\mathrm{t}\text{-}\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{c}\mathrm{r}\mathrm{e}\mathrm{t}\mathrm{e} basis.

Example 5.3　Michael space (see [36, Example 1.8.81]): regular quantifiable space with countable sn-nets, but not k-space. By Theorem 3.6, Michael spaces do not have \sigma \text{-}\mathrm{p}\mathrm{o}\mathrm{i}\mathrm{n}\mathrm{t}\text{-}\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{c}\mathrm{r}\mathrm{e}\mathrm{t}\mathrm{e} weak bases.

Example 5.4　There exists a weak first countable space with a \sigma-point-discrete wcs*-net, but not a \sigma-point-discrete cs*-net, see the text [35, Example 1.5.6], which gives the space X. This example shows that if the cs*-net in Problem 4.1 is reduced to a wcs*-net, the answer to the problem is negative.

Example 5.5　Sequential sector S_{\omega } (see [35, Example 1.5.21]): with \sigma-point-discrete cs-net and \sigma \text{-}\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}\mathrm{a}\mathrm{c}\mathrm{t}\text{-}\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{e} cs-net, but not \sigma \text{-}\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}\mathrm{a}\mathrm{c}\mathrm{t}\text{-}\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{e} sn-nets.

Example 5.6　Isbell-Mrowka space \psi (\mathbb{N}) (see [36, Example 1.8.41]): with \sigma-point-discrete nets, but not \sigma \text{-}\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}\mathrm{a}\mathrm{c}\mathrm{t}\text{-}\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{e} k-net.

References

[1] AleksandrovP SFedorchukV VZcevV I. The main aspects in the development of set-theoretic topology. Uspekhi Mat Nauk, 1978, 33(3): 3−48 (in Russian)

[2] Arhangel’skii A V. An addition theorem for the weight of sets lying in bicompacts. Dokl Akad Nauk SSSR 1959; 126(2): 239–241

[3] Arhangel’skii A V. Mappings and spaces. Russian Math Surveys 1966; 21(4): 115–162

[4] AullC ELowenR. Handbook of the History of General Topology, Vol 1. Dordrecht: Kluwer Academic Publishers, 1997

[5] Bing R H. Metrization of topological spaces. Canad J Math 1951; 3(2): 175–186

[6] Boone J R. Some characterizations of paracompactness in k-spaces. Fund Math 1971; 72(2): 145–153

[7] Burke D K, Davis S W. Spaces with σ-weakly hereditarily closure preserving base. Topology Proc 2010; 35: 9–18

[8] Burke D K, Engelking R, Lutzer D J. Hereditarily closure-preserving collections and metrization. Proc Amer Math Soc 1975; 51(2): 483–488

[9] BurkeD KLutzerD J. Recent advances in the theory of generalized metric spaces. In: Topology Proc 9th Ann Spring Topological Conf (Memphis State Univ, 1975). Lect Notes Pure and Appl Math 24. New York: Marcel Dekker Inc, 1976, 1–70

[10] FangJ LHausdorff. Biographies of world-famous mathematicians. Beijing: Science Press, 1995, 1240–1246

[11] Franklin S P. Spaces in which sequences suffice. Fund Math 1965; 57(1): 107–115

[12] Gao Z. ℵ-space is invariant under perfect mappings. Questions Answers Gen Topology 1987; 5(2): 271–279

[13] Ge X, Shen J, Ge Y. Spaces with σ-weakly hereditarily closure-preserving sn-networks. Novi Sad J Math 2007; 37(1): 33–37

[14] Ge Y. On sn-Metrizable spaces. Acta Math Sin Engl Ser 2002; 45(2): 355–360

[15] JiangJ G. On the metrizability of affine compactness and topological spaces, Acta Math

[16] GruenhageG. Generalized metric spaces. In: Handbook of Set-theoretic Topology (Kunen K, Vaughan J E, eds). Amsterdam: Elsevier Science Publishers B V, 1984, 423–501

[17] GruenhageG. Generalized metric spaces and metrization. In: Recent Progress in General Topology (Hušek M, Van Mill J, eds). Papers from the Prague Topological Symp (Prague, 1991). Amsterdam: Elsevier Science Publishers B V, 1992, 239–274

[18] GruenhageG. Metrizable spaces and generalizations. In: Recent Progress in General Topology II (Hušek M, Van Mill J, eds). Papers from the Prague Topological Symp (Prague, 2001). Amsterdam: Elsevier Science Publishers B V, 2002, 201–225

[19] Gruenhage G. A survey of D-spaces. Contemp Math 2011; 533: 13–28

[20] GruenhageG. Generalized metrizable spaces. In: Recent Progress in General Topology III (Hart K P, Van Mill J, Simon P, eds). Papers from the Prague Topological Symp (Prague, 2011). Amsterdam: Atlantis Press, 2014, 471–505

[21] Guthrie J A. A characterization of ℵ0-spaces. General Topology Appl 1971; 1(2): 105–110

[22] HartK PVan MillJSimonP. Recent Progress in General Topology III. Papers from the Prague Topological Symp (Prague, 2011). Amsterdam: Atlantis Press, 2014

[23] HartK PNagataJVaughanJ E. Encyclopedia of General Topology. Amsterdam: Elsevier Science Publishers B V, 2004

[24] HausdorffF. Grundzüge der Mengenlehre. Leipzig: Veriag von Veit und Comp, 1914 (in German)

[25] Lašnev N. Closed images of metric spaces. Dokl Akad Nauk SSSR 1966; 170: 505–507

[26] Lin F C. σ–point–discrete cs*–networks and wcs*–networks. J Adv Research Pure Math 2010; 2(3): 7–12

[27] LinF C. Topological algebra and generalized metric spaces. Xiamen: Xiamen Univ Press, 2012

[28] Lin F C, Lin S. Some notes on sequence-covering maps. J Math Res Appl 2014; 34(1): 97–104

[29] Lin F C, Lin S. Sequence-covering maps on generalized metric spaces. Houston J Math 2014; 40(3): 927–943

[30] Lin F C, Shen R X. Some notes on σ-point-discrete sn-networks. Adv Math (China) 2010; 39(2): 212–216

[31] Lin S. On normal separable ℵ-spaces. Questions Answers Gen Topology 1987; 5: 249–254

[32] Lin S. A study of pseudobases. Questions Answers Gen Topology 1988; 6: 81–97

[33] Lin S. On sequence-covering s-mappings. Adv Math (China) 1996; 25(6): 548–551

[34] Lin S. A note on the Arens’ space and sequential fan. Topology Appl 1997; 81(3): 185–196

[35] LinS. Point countable coverings and sequential coverings mappings. Beijing: Science Press, 2002

[36] LinS. Generalized metric spaces and mappings. Beijing: Science Press, 2007

[37] Lin S, Tanaka Y. Point-countable k-networks, closed maps, and related results. Topology Appl 1994; 59(1): 79–86

[38] Lin Y, Yan L. A note on spaces with a σ-compact-finite weak base. Tsukuba J Math 2004; 28(1): 85–91

[39] Liu C. Spaces with a σ-compact finite k-network. Questions Answers Gen Topology 1992; 10: 81–87

[40] Liu C. Notes on g-metrizable spaces. Topology Proc 2005; 29(1): 207–215

[41] Liu C. On weak bases. Topology Appl 2005; 150(13): 91–99

[42] Liu C, Lin S. On countable-to-one maps. Topology Appl 2007; 154(2): 449–454

[43] Liu C, Lin S, Ludwig L D. Spaces with a σ-point-discrete weak base. Tsukuba J Math 2008; 32(1): 165–177

[44] Liu C, Ludwig L D. Nagata-Smirnov revisited: spaces with σ-WHCP bases. Topology Proc 2005; 29(2): 559–565

[45] Michael E A. Another note on paracompact spaces. Proc Amer Math Soc 1957; 8(4): 822–828

[46] Michael E A. A quintuple quotient quest. General Topology Appl 1972; 2(2): 91–138

[47] MoritaKNagataJ. Topics in General Topology. Amsterdam: Elsevier Science Publishers B V, 1989

[48] Nagata J. On a necessary and sufficient condition of metrizability. J Inst Polytech Osaka City Univ 1950; 1(2A): 93–100

[49] Okuyama A. On a generalization of Σ-spaces. Pacific J Math 1972; 42(2): 485–495

[50] O’Meara P. On paracompactness in function spaces with the compact open topology. Proc Amer Math Soc 1971; 29(1): 183–189

[51] Shang Y, Wang S Z. The discussion of spaces with σ-WHCP or σ-compact-finite sets. J Capital Norm Univ Nat Sci Ed 2007; 28(6): 16–21

[52] Shen R, Lin S. Spaces with σ-point-discrete ℵ0-weak bases. Appl Math J Chinese Univ 2013; 28B(l): 116–126

[53] Sirois-Dumais R. Quasi- and weakly-quasi-firs seven-countable spaces. Topology Appl 1980; 11(2): 223–230

[54] Smirnov M. On the metrization of topological spaces. Uspekhi Mat Nauk 1951; 6(6): 100–111

[55] Urysohn P. Zurn Metrisationsproblem. Math Ann 1925; 94(1): 309–315

[56] Van DouwenE K. Simultaneous extension of continuous functions. PhD Thesis. Amsterdam: Free Univ of Amsterdam, 1975

[57] EditorialCommittee of the Encyclopedia of China. Encyclopedia of China (Mathematics Vol). Beijing: Encyclopedia of China Press, 1988

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