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Fucai Lin

Publications and source records attributed to Fucai Lin.

At least 19 recordsLinked to original sources
arXiv2026

Characterizations of quotient spaces for Lindel\"of strongly topological gyrogroups

Let $\mathscr{L}$ be the class of Lindel\"of spaces such that $\mathscr{L}$ is closed under finite products. In this paper, we prove that if $G \in \mathscr{L}$ is a strongly topological gyrogroup, then $G$ is range-metrizable. Furthermore, we prove that if $H$ is a strong subgyrogroup of a strongly topological gyrogroup $G \in \mathscr{L}$, then every compact $G_\delta$-set in the quotient space $G/H$ is Dugundji. Finally, for any strongly topological gyrogroup $G$ and any closed strong subgyrogroup $N$ of $G$, if $G \in \mathscr{L}$ and the quotient space $G/N$ is locally compact, then the inequality $w(G/N) \leq c$ is equivalent to the separability of $G/N$. Our results extend the classical results from topological groups to the class of strongly topological gyrogroups in the literature.

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arXiv2026

The separability embedding of $\sigma$-compact strongly topological gyrogroups

In this paper, it is shown that every right $\omega$-narrow strongly topological gyrogroup $G$ is right $\omega$-balanced by applying the gyrosemidirect product groups. Then we investigate the class of $\sigma$-compact strongly topological gyrogroups, and conclude that every $\sigma$-compact strongly topological gyrogroup is range-metrizable. By applying these results, we discuss the separability embedding of $\sigma$-compact strongly topological gyrogroups, and claim that the following three statements (a)-(c) are equivalent for any $\sigma$-compact strongly topological gyrogroup $G$: \smallskip (a) $G$ is homeomorphic to a subspace of a separable regular space; \smallskip (b) $G$ is topologically gyrogroup isomorphic to a subgyrogroup of a separable strongly topological gyrogroup; \smallskip (c) $G$ is topologically gyrogroup isomorphic to a closed subgyrogroup of a separable path-connected, locally path-connected strongly topological gyrogroup. The above results extend the classical results from topological groups to the class of strongly topological gyrogroups in the literature.

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arXiv2025

Suitable sets for topological groups revisited

A discrete subset $S$ of a topological group $G$ is called a {\it suitable set} for $G$ if $S\cup \{e\}$ is closed in $G$ and the subgroup generated by $S$ is dense in $G$, where $e$ is the identity element of $G$. In this paper, the existence of suitable sets in topological groups is studied. It is proved that, for a non-separable $k_{\omega}$-space $X$ without non-trivial convergent sequences, the $snf$-countability of $A(X)$ implies that $A(X)$ does not have a suitable set, which gives a partial answer to \cite[Problem 2.1]{TKA1997}. Moreover, the existence of suitable sets in some particular classes of linearly orderable topological groups is considered, where Theorem~\ref{t4} provides an affirmative answer to \cite[Problem 4.3]{ST2002}. Then, topological groups with an $\omega^{\omega}$-base are discussed, and every linearly orderable topological group with an $\omega^{\omega}$-base being metrizable is proved; thus it has a suitable set. Further, it follows that each topological group $G$ with an $\omega^{\omega}$-base has a suitable set whenever $G$ is a $k$-space, which gives a generalization of a well-known result in \cite{CM}. Finally, some cardinal invariant of topological groups with a suitable set are provided. Some results of this paper give some partial answers to some open problems posed in~\cite{DTA} and~\cite{TKA1997} respectively.

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arXiv2025

The existence of suitable sets in locally compact strongly topological gyrogroups

A subset $S$ of a topological gyrogroup $G$ is said to be a {\it suitable set} for $G$ if $S$ is discrete, the gyrogroup generated by $S$ is dense in $G$, and $S\cup \{0\}$ is closed in $G$, where $0$ is the identity element of $G$. In this paper, it is proved that every locally compact strongly topological gyrogroup has a suitable set, which gives an affirmative answer to a question posed by F. Lin, et al. in \cite{key14}.

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arXiv2025

Strongly topologically orderable gyrogroups with a suitable set

A discrete subset $S$ of a topologically gyrogroup $G$ is called a {\it suitable set} for $G$ if $S\cup \{1\}$ is closed and the subgyrogroup generated by $S$ is dense in $G$, where $1$ is the identity element of $G$. In this paper, we mainly study the existence of suitable set of strongly topologically orderable gyrogroups, which extends some result in some papers in the literature. In particular, the existences of suitable set of each locally compact or not totally disconnected strongly topologically orderable gyrogroup are affirmative.

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arXiv2025

The characterizations of hyperspaces and free topological groups with an $\omega^\omega$-base

A topological space $(X, \tau)$ is said to be have an {\it $\omega^\omega$-base} if for each point $x\in X$ there exists a neighborhood base $\{U_{\alpha}[x]: \alpha\in\omega^\omega\}$ such that $U_{\beta}[x]\subset U_{\alpha}[x]$ for all $\alpha\leq\beta$ in $\omega^\omega$. In this paper, the characterization of a space $X$ is given such that the free Abelian topological group $A(X)$, the hyperspace $CL(X)$ with the Vietoris topology and the hyperspace $CL(X)$ with the Fell topology have $\omega^\omega$-bases respectively. The main results are listed as follows: (1) For a Tychonoff space $X$, the free Abelian topological group $A(X)$ is a $k$-space with an $\omega^\omega$-base if and only if $X$ is a topological sum of a discrete space and a submetrizable $k_\omega$-space. (2) If $X$ is a metrizable space, then $(CL(X), \tau_V)$ has an $\omega^\omega$-base if and only if $X$ is separable and the boundary of each closed subset of $X$ is $\sigma$-compact. (3) If $X$ is a metrizable space, then $(CL(X), \tau_F)$ has an $\omega^\omega$-base consisting of basic neighborhoods if and only if $X$ is a Polish space. (4) If $X$ is a metrizable space, then $(CL(X), \tau_F)$ is a Fr\'echet-Urysohn space with an $\omega^\omega$-base, if and only if $(CL(X), \tau_F)$ is first-countable, if and only if $X$ is a locally compact and second countable space.

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arXiv2024

Some generalized metric properties of $n$-semitopological groups

A semitopological group $G$ is called {\it an $n$-semitopological group}, if for any $g\in G$ with $e\not\in\overline{\{g\}}$ there is a neighborhood $W$ of $e$ such that $g\not\in W^{n}$, where $n\in\mathbb{N}$. The class of $n$-semitopological groups ($n\geq 2$) contains the class of paratopological groups and Hausdorff quasi-topological groups. Fix any $n\in\mathbb{N}$. Some properties of $n$-semitopological groups are studied, and some questions about $n$-semitopological groups are posed. Some generalized metric properties of $n$-semitopological groups are discussed, which contains mainly results are that (1) each Hausdorff first-countable 2-semitopological group admits a coarsersemi-metrizable topology; (2) each locally compact, Baire and $\sigma$-compact 2-semitopological group is a topological group; (3) the condensation of some kind of 2-semitopological groups topologies are given. Finally, some cardinal invariants of $n$-semitopological groups are discussed.

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arXiv2023

A note on complementary knowledge spaces

The pair $(Q, \mathscr{K})$ is a {\it knowledge space} if $\bigcup\mathscr{K}=Q$ and $\mathscr{K}$ is closed under union, where $Q$ is a nonempty set and $\mathscr{K}$ is a family of subsets of $Q$. A knowledge space $(Q, \mathscr{K})$ is called {\it complementary} if there exists a non-discrete knowledge space $(Q, \mathscr{L})$ such that the following (i) and (ii) satisfy: (i) for any $q\in Q$, there are finitely many $K_{1}, \cdots, K_{n}\in \mathscr{K}$ and $L_{1}, \cdots, L_{m}\in \mathscr{L}$ such that $$(\bigcap_{i=1}^{n}K_{i})\cap (\bigcap_{j=1}^{m}L_{j})=\{q\};$$ (ii) $\mathscr{K}\cap \mathscr{L}=\{\emptyset, Q\}$. In this paper, the existence of a complementary knowledge space for each knowledge space is proved, and a method of the construction of complementary finite knowledge spaces is given.

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arXiv2023

The quasi-metrizability of hyperspaces

For a space $X$, let $(CL(X), \tau_V)$, $(CL(X), \tau_{locfin})$ and $(CL(X), \tau_F)$ be the set $CL(X)$ of all nonempty closed subsets of $X$ which are endowed with Vietoris topology, locally finite topology and Fell topology respectively. We prove that $(CL(X), \tau_V)$ is quasi-metrizable if and only if $X$ is a separable metrizable space and the set of all non-isolated points of $X$ is compact, $(CL(X), \tau_{locfin})$ is quasi-metrizable or symmetrizable if and only if $X$ is metrizable and the set of all non-isolated points of $X$ is compact, and $(CL(X), \tau_F)$ is quasi-metrizable if and only if $X$ is hemicompact and metrizable. As an application, we give a negative answer to a Conjecture in \cite{LL2022}.

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arXiv2023

The characterizations of dense-pseudocompact and dense-connected spaces

Assume that $\mathcal{P}$ is a topological property of a space $X$, then we say that $X$ is {\it dense-$\mathcal{P}$} if each dense subset of $X$ has the property $\mathcal{P}$. In this paper, we mainly discuss dense subsets of a space $X$, and we prove that: (1) if $X$ is Tychonoff space, then $X$ is dense-pseudocompact iff the range of each continuous real-valued function $f$ on $X$ is finite, iff $X$ is finite, iff $X$ is hereditarily pseudocompact; (2) $X$ is dense-connected iff $\overline{U}=X$ for any non-empty open subset $U$ of $X$; (3) $X$ is dense-ultraconnected iff for point $x\in X$, we have $\overline{\{x\}}=X$ or $\{x\}\cup (X\setminus\overline{\{x\}})$ is the unique open neighborhood of $x$ in $\{x\}\cup (X\setminus\overline{\{x\}})$, iff for any two points $x$ and $y$ in $X$, we have $x\in \overline{\{y\}}$ or $y\in \overline{\{x\}}$. Moreover, we give a characterization of a topological group (resp., paratopological group, quasi-topological group) $G$ such that $G$ is dense-connected.

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arXiv2022

Dense-separable groups and its applications in $d$-independence

A topological space is called {\it dense-separable} if each dense subset of its is separable. Therefore, each dense-separable space is separable. We establish some basic properties of dense-separable topological groups. We prove that each separable space with a countable tightness is dense-separable, and give a dense-separable topological group which is not hereditarily separable. We also prove that, for a Hausdorff locally compact group , it is locally dense-separable iff it is metrizable. Moreover, we study dense-subgroup-separable topological groups. We prove that, for each compact torsion (or divisible, or torsion-free, or totally disconnected) abelian group, it is dense-subgroup-separable iff it is dense-separable iff it is metrizable. Finally, we discuss some applications in $d$-independent topological groups and related structures. We prove that each regular dense-subgroup-separable abelian semitopological group with $r_{0}(G)\geq\mathfrak{c}$ is $d$-independent. We also prove that, for each regular dense-subgroup-separable bounded paratopological abelian group $G$ with $|G|>1$, it is $d$-independent iff it is a nontrivial $M$-group iff each nontrivial primary component $G_{p}$ of $G$ is $d$-independent. Apply this result, we prove that a separable metrizable almost torsion-free paratopological abelian group $G$ with $|G|=\mathfrak{c}$ is $d$-independent. Further, we prove that each dense-subgroup-separable MAP abelian group with a nontrivial connected component is also $d$-independent.

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arXiv2022

Hyperspaces with a countable character of closed subsets

For a regular space $X$, the hyperspace $(CL(X), \tau_{F})$ (resp., $(CL(X), \tau_{V})$) is the space of all nonempty closed subsets of $X$ with the Fell topology (resp., Vietoris topology). In this paper, we give the characterization of the space $X$ such that the hyperspace $(CL(X), \tau_{F})$ (resp., $(CL(X), \tau_{V})$) with a countable character of closed subsets. We mainly prove that $(CL(X), \tau_F)$ has a countable character on each closed subset if and only if $X$ is compact metrizable, and $(CL(X), \tau_F)$ has a countable character on each compact subset if and only if $X$ is locally compact and separable metrizable. Moreover, we prove that $(\mathcal{K}(X), \tau_V)$ have the compact-$G_\delta$ property if and only if $X$ have the compact-$G_\delta$ property and every compact subset of $X$ is metrizable.

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arXiv2022

Some properties of Pre-topological groups

In this paper, we pose the concepts of pre-topological groups and some generalizations of pre-topological groups. First, we systematically investigate some basic properties of pre-topological groups; in particular, we prove that each $T_{0}$ pre-topological group is regular and every almost topological group is completely regular which extends A.A. Markov's theorem to the class of almost topological groups. Moreover, it is shown that an almost topological group is $\tau$-narrow if and only if it can be embedded as a subgroup of a pre-topological product of almost topological groups of weight less than or equal to $\tau$. Finally, the cardinal invariant, the precompactness and the resolvability are investigated in the class of pre-topological groups.

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arXiv2022

Some properties of Pre-uniform spaces

In this paper, we introduce the notions of pre-uniform spaces and pre-proximities and investigate some basic properties about them, where the definition of pre-uniformity here is different with the pre-uniformities which are studied in \cite{BR2016}, \cite{GM2007} and \cite{K2016} respectively. First, we prove that each pre-uniform pre-topology is regular, and give an example to show that there exists a pre-uniform structure on a finite set such that the pre-uniform pre-topology is not discrete. Moreover, we give three methods of generating (strongly) pre-uniformities, that is, the definition of a pre-base, a family of strongly pre-uniform covers, or a family of strongly pre-uniform pseudometrics. As an application, we show that each strongly pre-topological group is completely regular. Finally, we pose the concept of the pre-proximity on a set and discuss some properties of the pre-proximity.

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arXiv2021

A note on knowledge structures delineated by fuzzy skill multimaps

Fuzzy skill multimaps can describe individuals' knowledge states from the perspective of latent cognitive abilities. The significance of discriminative knowledge structure is reducing repeated testing and the workload for students, which responds to the so-called `double reduction policy'. As an application in knowledge assessment, the bi-discriminative knowledge structure is useful for evaluating `relative independence' items or knowledge points. Moreover, it is of great significance that which kind of fuzzy skill multimap delineates a discriminative or bi-discriminative knowledge structure. Moreover, we give a characterization of fuzzy skill multimaps such that the delineated knowledge structures are knowledge spaces, learning spaces and simple closure spaces respectively. Further, there are some interesting applications in the meshing of the delineated knowledge structures and the distributed fuzzy skill multimaps. We also give a more precise analysis with regard to the separability (resp. bi-separability) on which condition the information collected within a local assessment can reflect an assessment at the global level, and which condition any of the local domains can localize from given global assessment.

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arXiv2021

The language of pre-topology in knowledge spaces

We systematically study some basic properties of the theory of pre-topological spaces, such as, pre-base, subspace, axioms of separation, connectedness, etc. Pre-topology is also known as knowledge space in the theory of knowledge structures. We discuss the language of axioms of separation of pre-topology in the theory of knowledge spaces, the relation of Alexandroff spaces and quasi ordinal spaces, and the applications of the density of pre-topological spaces in primary items for knowledge spaces. In particular, we give a characterization of a skill multimap such that the delineate knowledge structure is a knowledge space, which gives an answer to a problem in \cite{falmagne2011learning} or \cite{XGLJ} whenever each item with finitely many competencies; moreover, we give an algorithm to find the set of atom primary items for any finite knowledge spaces.

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arXiv2021

A note on hyperspaces by closed sets with Vietoris topology

For a topological space $X$, let $CL(X)$ be the set of all non-empty closed subset of $X$, and denote the set $CL(X)$ with the Vietoris topology by $(CL(X), \mathbb{V})$. In this paper, we mainly discuss the hyperspace $(CL(X), \mathbb{V})$ when $X$ is an infinite countable discrete space. As an application, we first prove that the hyperspace with the Vietoris topology on an infinite countable discrete space contains a closed copy of $n$-th power of Sorgenfrey line for each $n\in\mathbb{N}$. Then we investigate the tightness of the hyperspace $(CL(X), \mathbb{V})$, and prove that the tightness of $(CL(X), \mathbb{V})$ is equal to the set-tightness of $X$. Moreover, we extend some results about the generalized metric properties on the hyperspace $(CL(X), \mathbb{V})$. Finally, we give a characterization of $X$ such that $(CL(X), \mathbb{V})$ is a $\gamma$-space.

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