Fucai Lin

MinNan Normal University · School of Mathematics and Statistics

The theory of knowledge spaces (KST) which is regarded as a mathematical framework for the assessment of knowledge and advices for further learning. Now the theory of knowledge spaces has many applications in education. From the topological point of view, we discuss the language of the theory of knowledge spaces by the axioms of separation and the...

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 t...

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 from the preuniformities which are studied in [1], [8] and [12], respectively. First, we prove that each pre-uniform pre-topology is regular, and give an example...

Fuzzy skill functions connect knowledge states at the performance level with latent cognitive abilities at the competence level. Given that there may exist precedence relations among skills, the main idea of this study is trying to develop fuzzy competence structures restricted on the fuzzy sets of skills that are likely to occur. We consider the k...

The dichotomous knowledge structure theory was introduced to polytomous items, which is proved meaningful in representing individuals' partial mastery of items. Given that different proficiencies in skills may be required to solve polytomous items, fuzzy skills can be applied to represent latent cognitive abilities of individuals. Then we consider...

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),V). In this paper, we mainly discuss the hyperspace (CL(X),V) when X is an infinite countable discrete space. As an application, we first prove that the hyperspace with the Vietoris topology on an infini...

In this paper, we introduce the notions of pre-uniform spaces and pre-proximities and investigate some basic properties about them. 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...

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....

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-discrimin...

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 rela...

A space X is submaximal if any dense subset of X is open. In this paper, we prove that every submaximal topological gyrogroup of non-measurable cardinality is strongly σ-discrete. Moreover, we prove that every submaximal strongly topological gyrogroup of non-measurable cardinality is hereditarily paracompact.

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...

In this paper, it is proved that every topological gyrogroup $G$ is topologically groupoid isomorphic to a closed subgyrogroup of a connected, locally connected topological gyrogroup $G^{\bullet}$.

A subset S of a paratopological group G is a suitable set for G, if S is a discrete subspace of G, \(S\cup \{e\}\) is closed, and the subgroup \(\langle S\rangle \) of G generated by S is dense in G. Suitable sets in topological groups were studied by many authors. The aim of the present paper is to provide a start-up for a general investigation of...

The symbol S( X ) denotes the hyperspace of finite unions of convergent sequences in a Hausdor˛ space X . This hyper-space is endowed with the Vietoris topology. First of all, we give a characterization of convergent sequence in S( X ). Then we consider some cardinal invariants on S( X ), and compare the character, the pseudocharacter, the sn-chara...

Some results between the properties of strongly topological gyrogroups and the properties of their remainders are established. In particular, if a strongly topological gyrogroup G is non-locally compact and G has a first-countable remainder, then \(\chi (G)\le \omega _{1}\), \(\omega (G)\le 2^{\omega }\) and \(|bG|\le 2^{\omega _{1}}\). Moreover, i...

Two non-discrete Hausdorff group topologies τ and δ on a group G are called transversal if the least upper bound τ ⋁ δ of τ and δ is the discrete topology. In this paper, we discuss the existence of transversal group topologies on locally pseudocompact, locally precompact, or locally compact groups. We prove that each locally pseudocompact, connect...

Let (U, R) be an approximation space with U being non-empty set and R being an equivalence relation on U, and let G¯ and G̲ be the upper approximation and the lower approximation of subset G of U. A topological rough group G is a rough group G=(G̲,G¯) endowed with a topology, which is induced from the upper approximation space G¯, such that the pro...

We discuss the class of paratopological groups which admits a transversal, T1-independent and T1-complementary paratopological group topology. We show that the Sorgenfrey line does not admit a T1-complementary Hausdorff paratopological group topology, which gives a negative answer to [5, Problem 10]. We give a very useful criterion for transversali...

The concept of gyrogroups, with a weaker algebraic structure without associative law, was introduced under the background of c-ball of relativistically admissible velocities with the Einstein velocity addition. A topological gyrogroup is just a gyrogroup endowed with a compatible topology such that the multiplication is jointly continuous and the i...

A space $X$ is submaximal if any dense subset of $X$ is open. In this paper, we prove that every submaximal topological gyrogroup of non-measurable cardinality is strongly $\sigma$-discrete. Moreover, we prove that every submaximal strongly topological gyrogroup of non-measurable cardinality is hereditarily paracompact.

A discrete subset $S$ of a topological gyrogroup $G$ with the identity $0$ is said to be a {\it suitable set} for $G$ if it generates a dense subgyrogroup of $G$ and $S\cup \{0\}$ is closed in $G$. In this paper, it was proved that each countable Hausdorff topological gyrogroup has a suitable set; moreover, it is shown that each separable metrizabl...

A paratopological group $G$ has a {\it suitable set} $S$. The latter means that $S$ is a discrete subspace of $G$, $S\cup \{e\}$ is closed, and the subgroup $\langle S\rangle$ of $G$ generated by $S$ is dense in $G$. Suitable sets in topological groups were studied by many authors. The aim of the present paper is to provide a start-up for a general...

The symbol $\mathcal{S}(X)$ denotes the hyperspace of finite unions of convergent sequences in a Hausdorff space $X$. This hyperspace is endowed with the Vietoris topology. First of all, we give a characterization of convergent sequence in $\mathcal{S}(X)$. Then we consider some cardinal invariants on $\mathcal{S}(X)$, and compare the character, th...

The free topological vector space V(X) over a Tychonoff space X is a pair consisting of a topological vector space V(X) and a continuous mapping i=iX:X→V(X) such that every continuous mapping f from X to a topological vector space E gives rise to a unique continuous linear operator f‾:V(X)→E with f=f‾∘i. In this paper, the k-property, Fréchet-Uryso...

The concept of gyrogroups, with a weaker algebraic structure without associative law, was introduced under the background of $c$-ball of relativistically admissible velocities with Einstein velocity addition. The class of topological gyrogroups is just the gyrogroups endowed with a compatible topology such that the multiplication is jointly continu...

Topological gyrogroups, with a weaker algebraic structure without associative law, have been investigated recently. We prove that each $T_{0}$-strongly topological gyrogroup is completely regular. We also prove that every $T_{0}$-strongly topological gyrogroup with a countable pseudocharacter is submetrizable. Finally, we prove that the left coset...

Let $(U, R)$ be an approximation space with $U$ being non-empty set and $R$ being an equivalence relation on $U$, and let $\overline{G}$ and $\underline{G}$ be the upper approximation and the lower approximation of subset $G$ of $U$. A topological rough group $G$ is a rough group $G=(\underline{G}, \overline{G})$ endowed with a topology, which is i...

In this paper, we discuss some properties of of $G$-hull, $G$-kernel and $G$-connectedness, and extend some results of \cite{life34}. In particular, we prove that the $G$-connectedness are preserved by countable product. Moreover, we introduce the concept of $G$-topological group, and prove that a $G$-topological group is a $G$-topology under the a...

Topological gyrogroups, with a weaker algebraic structure than groups, have been investigated recently. In this paper, we prove that every feathered strongly topological gyrogroup is paracompact, which implies that every feathered strongly topological gyrogroup is a $D$-space and gives partial answers to two questions posed by A.V.Arhangel' ski\v\i...

We study the heredity of the classes of generalized metric spaces (for example, spaces with a $\sigma$-hereditarily closure-preserving $k$-network, spaces with a point-countable base, spaces with a base of countable order, spaces with a point-regular base, Nagata-spaces, $c$-semi-stratifiable spaces, $\gamma$-spaces, semi-metrizable spaces) to the...

The free topological vector space V (X) over a Tychonoff space X is a pair consisting of a topological vector space V (X) and a continuous mapping i = i X : X → V (X) such that every continuous mapping f from X to a topological vector space E gives rise to a unique continuous linear operator f : V (X) → E with f = f • i. In this paper, the k-proper...

We discuss the class of paratopological groups which admits a transversal, $T_{1}$-independent and $T_{1}$-complementary paratopological group topology. We show that the Sorgenfrey line does not admit a $T_{1}$-complementary Hausdorff paratopological group topology, which gives a negative answer to \cite[Problem 10]{AT2017}. We give a very useful c...

Two non-discrete Hausdorff group topologies $\tau, \delta$ on a group $G$ are called {\it transversal} if the least upper bound $\tau\vee \delta$ of $\tau$ and $\delta$ is the discrete topology. In this paper, we main discuss the transversality of locally pseudocompact, locally precompact or locally compact groups. We prove that each locally pseudo...

In this paper, we give an affirmative answer to Yamada's Conjecture on free topological groups, which was posed in [K. Yamada, {\it Fréchet-Urysohn spaces in free topological groups}, Proc. Amer. Math. Soc., {\bf 130}(2002), 2461--2469.].

In this paper, we discuss some properties of of G-hull, G-kernel and G-connectedness, and extend some results of [28]. In particular, we prove that the G-connectedness are preserved by countable product. Moreover, we introduce the concept of G-topological group, and prove that a G-topological group is a G-topology under the assumption of the regula...

Topological gyrogroups, with a weaker algebraic structure than groups, have been investigated recently. In this paper, we prove that every feathered strongly topological gyrogroup is paracompact, which implies that every feathered strongly topological gyrogroup is a D-space and gives partial answers to two questions posed by A.V. Arhangel?ski?(2010...

The $H$-space, denoted as $(\mathbb{R}, \tau_{A})$, has $\mathbb{R}$ as its point set and a basis consisting of usual $\mathbb{R}$ neighborhood at points of $A$ while taking Sorgenfrey neighborhoods at points of $\mathbb{R}-A$. In this paper, we mainly discuss some topological properties of $H$-spaces, such as zero-dimension, local compactness, $\s...

In this paper, we give an affirmative answer to Yamada's Conjecture on free topological groups, which was posed in [K. Yamada, {\it Fr\'echet-Urysohn spaces in free topological groups}, Proc. Amer. Math. Soc., {\bf 130}(2002), 2461--2469.].

A space X is called a kR-space, if X is Tychonoff and the necessary and sufficient condition for a real-valued function f on X to be continuous is that the restriction of f to each compact subset is continuous. In this paper, we discuss the kR-property of products of sequential fans and free Abelian topological groups by applying the κ-fan introduc...

The multiplication of a semitopological (quasitopological) group G is called sequentially continuous if the product map of G×G into G is sequentially continuous. In this paper, we mainly consider the properties of semitopological (quasitopological) groups with sequentially continuous multiplications and three-space problems in quasitopological grou...

In this paper, we continue the study of the symmetric products of generalized metric spaces in [39]. We consider the topological properties P such that the n-fold symmetric product Fn(X) of a topological space X has the topological properties P if and only if the space X or the product Xn does for each or some n∈N. Depending on the operations under...

Given a Tychonoff space X, let F(X) and A(X) be respectively the free topological group and the free Abelian topological group over X in the sense of Markov. For every n∈N, let Fn(X) (resp. An(X)) denote the subspace of F(X) (resp. A(X)) that consists of words of reduced length at most n with respect to the free basis X. In this paper, we mainly di...

The free topological vector space $V(X)$ over a Tychonoff space $X$ is a pair consisting of a topological vector space $V(X)$ and a continuous map $i=i_{X}: X\rightarrow V(X)$ such that every continuous mapping $f$ from $X$ to a topological vector space $E$ gives rise to a unique continuous linear operator $\overline{f}: V(X)\rightarrow E$ with $f=...

A space $X$ is of countable type (resp. subcountable type) if every compact subspace $F$ of $X$ is contained in a compact subspace $K$ that is of countable character (resp. countable pseudocharacter) in $X$. In this paper, we mainly show that: (1) For a functionally Hausdorff space $X$, the free paratopological group $FP(X)$ and the free abelian pa...

A space $X$ is called a $k_{R}$-space, if $X$ is Tychonoff and the necessary and sufficient condition for a real-valued function $f$ on $X$ to be continuous is that the restriction of $f$ on each compact subset is continuous. In this paper, we shall discuss the $k_{R}$-property on the sequential fans, and show the following two results: (1) the spa...

Given a Tychonoff space $X$, let $A(X)$ be the free Abelian topological group over $X$ in the sense of Markov. For every $n\in\mathbb{N}$, let $A_n(X)$ denote the subspace of $A(X)$ that consists of words of reduced length at most $n$ with respect to the free basis $X$. In this paper, we show that $A_4(X)$ is a $k$-space if and only if $A(X)$ is a...

Given an increasing sequence (X n) n∈ω of quasi-uniform spaces and paratopological groups, we study the topology of the direct limits qu-lim −→ X n and pg-lim −→ X n of the sequence (X n) n∈ω in the categories of quasi-uniform spaces and paratopological groups, respectively. First, we prove that the quasi-uniformity of the quasi-uniform direct limi...

Given a Tychonoff space $X$, let $F(X)$ and $A(X)$ be respectively the free topological group and the free Abelian topological group over $X$ in the sense of Markov. In this paper, we discuss two topological properties in $F(X)$ or $A(X)$, namely the countable tightness and $\mathfrak G$-base. We provide some characterizations of the countable tigh...

In this paper, we study statistical versions of sequential and Fréchet-Urysohn. Some interesting properties of these spaces are obtained.

A topological space G is said to be a rectifiable space provided that there are a surjective homeomorphism ϕ:G×G→G×G and an element e∈G such that π1oϕ=π1 and for every x∈G, ϕ(x, x)=(x, e), where π1:G×G→G is the projection to the first coordinate. In this paper, we first prove that each locally compact rectifiable space is paracompact, which gives a...

In this addendum we give an example to show that there is an error in the proof of Lemma 4.2 in “ and on free topological groups” [Topol. Appl. 176 (2014) 10–21].

In this paper, we mainly discuss the class of charming spaces, which was introduced by A.V. Arhangel'skii in [Remainders of metrizable spaces and a generalization of Lindel\"{o}f $\Sigma$-spaces, Fund. Math., 215(2011), 87-100]. First, we show that there exists a charming space $X$ such that $X^{2}$ is not a charming space. Then we discuss some pro...

Let $FP(X)$ be the free paratopological group over a topological space $X$. For each non-negative integer $n\in\mathbb{N}$, denote by $FP_{n}(X)$ the subset of $FP(X)$ consisting of all words of reduced length at most $n$, and $i_{n}$ by the natural mapping from $(X\bigoplus X^{-1}\bigoplus\{e\})^{n}$ to $FP_{n}(X)$. In this paper, we mainly improv...

Given a Tychonoff space $X$, let $F(X)$ and $A(X)$ be respectively the free topological group and the free Abelian topological group over $X$ in the sense of Markov. In this paper, we provide some topological properties of $X$ whenever one of $F(X)$, $A(X)$ and the finite levels of $F(X)$ and $A(X)$ is $q$-space (in particular, locally $\omega$-bou...

A topological space $G$ is said to be a {\it rectifiable space} provided that there are a surjective homeomorphism $\varphi :G\times G\rightarrow G\times G$ and an element $e\in G$ such that $\pi_{1}\circ \varphi =\pi_{1}$ and for every $x\in G$, $\varphi (x, x)=(x, e)$, where $\pi_{1}: G\times G\rightarrow G$ is the projection to the first coordin...

Given a Tychonoff space $X$, let $F(X)$ and $A(X)$ be respectively the free topological group and the free Abelian topological group over $X$ in the sense of Markov. For every $n\in\mathbb{N}$, let $F_{n}(X)$ (resp. $A_n(X)$) denote the subspace of $F(X)$ (resp. $A(X)$) that consists of words of reduced length at most $n$ with respect to the free b...

Let X be a metrizable space. Let FP(Y) and AP(X) be the free paratopological group over X and the free Abelian paratopological group over X, respectively. Firstly, we use asymmetric locally convex spaces to prove that if Y is a subspace of X then AP(Y) is topological subgroup of AP(X). Then, we mainly prove that: (a) if the tightness of AP(X) is co...

In this paper, we investigate certain networks on free topological groups and free Abelian topological groups and obtain the following: (1) is a submetrizable, σ-space if and only if X is a submetrizable, σ-space; (2) Let X be a -metrizable, μ-space. is a k-space if and only if X is discrete or a k-space with a countable k-network consisting of com...

In this paper sequentially compact sets, weakly first-countable sets and generalized metric sets in extensions of topological groups are studied. Some three space properties on convergence phenomena are obtained. It is shown that (1) if H is a closed subgroup of a topological group G such that all sequentially compact subsets of both the group H an...

In this paper, we investigate copies of S-omega and S-2 on free topological groups. By applying these results, we show that, for a paracompact space with a point-countable k-network, X is discrete or compact if F-5(X) is Frechet-Urysohn, which generalizes Yamada's theorem (Yamada [26]). We also give a negative answer to Yamada's conjecture (Yamada...

We mainly discuss the remainders of Hausdorff compactifications of paratopological groups or semitopological groups. Thus, we show that if a nonlocally compact semitopological group G has a compactification bG such that the remainder Y = bG \ G possesses a locally countable network, then G has a countable π -character and is also first-countable, t...

In this paper, we describe in internal terms the kernel of the canonical homomorphism of a semitopological group G onto the -reflection of G, which answers a problem of M. Tkachenko.

Let G be a paratopological group. Then G is said to be pseudobounded (resp. ω-pseudobounded) if for every neighbourhood V of the identity e in G, there exists a natural number n such that G = Vn (resp.we have G = ∪ n∈N Vn). We show that every feebly compact (2-pseudocompact) pseudobounded (ω-pseudobounded) premeager paratopological group is a topol...

In this short article, we give complete answers to four problems of Arhangel'skii on diagonal-flexible spaces or rotoids. We list the four problems as follows: Problem A. Is every compact diagonal-flexible space rectifiable? Problem B. Is every diagonal-flexible space with a countable π-character first-countable? Problem C. Is every diagonal-flexib...

A topological space G is said to be a rectifiable space provided that there are a surjective homeomorphism φ:G×G→G×Gφ:G×G→G×G and an element e∈Ge∈G such that π1∘φ=π1π1∘φ=π1 and for every x∈Gx∈G we have φ(x,x)=(x,e)φ(x,x)=(x,e), where π1:G×G→Gπ1:G×G→G is the projection to the first coordinate. We firstly define the concept of rectifiable completion...

In this paper, we mainly discuss the cardinal invariants on some class of paratopological groups. For each i∈{0,1,2,3,3.5}i∈{0,1,2,3,3.5}, we define the class of locally TiTi-minimal paratopological groups by the conditions that, for a TiTi paratopological group (G,τ)(G,τ), there exists a τ-neighborhood U of the neutral element such that U fails to...

A topological space G is said to be a rectifiable space provided that there are a surjective homeomorphism φ:G×G→G×G and an element e∈G such that π 1∘φ=π 1 and for every x∈G we have φ(x,x)=(x,e), where π 1:G×G→G is the projection to the first coordinate. Let G be a rectifiable space and C(G) be the family of all non-empty compact subsets of G. In t...

A space X is a rotoid if there are a special point e∈Xe∈X and a homeomorphism H from X2X2 onto itself such that H(x,x)=(x,e)H(x,x)=(x,e) and H(e,x)=(e,x)H(e,x)=(e,x) for each x∈Xx∈X. Rotoids are generalizations of topological groups, and the Sorgenfrey line is a rotoid and not a topological group. In this paper, we prove some cardinal invariants fo...

A space X is said to be weakly quasi-first-countable if and only if for all x ∈ X, there exists countably many countable families of decreasing subsets containing x such that a set O is open if and only if for any x ∈ O, O contains a member of each family associated to x. For a space X, we denote the countable σ-product of X endowed with the box to...

A rectifiable space (or a paratopological group) G is compactly generated if G = 〈K〉 for some compact subset K of G. In this paper, we mainly discuss compactly generated rectifiable spaces or paratopological groups. The main results are that: (1) each σ-compact metrizable rectifiable space containing a dense compactly generated rectifiable subspace...

In this paper, we firstly construct a Hausdorff non-submetrizable paratopological group $G$ in which every point is a $G_{\delta}$-set, which gives a negative answer to Arhangel'ski\v{\i}\ and Tkachenko's question [Topological Groups and Related Structures, Atlantis Press and World Sci., 2008]. We prove that each first-countable Abelian paratopolog...

A space $X$ is said to be $\pi$-metrizable if it has a $\sigma$-discrete $\pi$-base. In this paper, we mainly give affirmative answers for two questions about $\pi$-metrizable spaces. The main results are that: (1) A space $X$ is $\pi$-metrizable if and only if $X$ has a $\sigma$-hereditarily closure-preserving $\pi$-base; (2) $X$ is $\pi$-metrizab...

In this paper, we mainly discuss some generalized metric properties and the character of the free paratopological groups, and extend several results valid for free topological groups to free paratopological groups.

Suppose that $X$ is a subspace of a Tychonoff space $Y$. Then the embedding mapping $e_{X, Y}: X\rightarrow Y$ can be extended to a continuous monomorphism $\hat{e}_{X, Y}: AP(X)\rightarrow AP(Y)$, where $AP(X)$ and $AP(Y)$ are the free Abelian paratopological groups over $X$ and $Y$, respectively. In this paper, we mainly discuss when $\hat{e}_{X,...

We mainly introduce some weak versions of the $M_{1}$-spaces, and study some properties about these spaces. The mainly results are that: (1) If $X$ is a compact scattered space and $i(X)\leq 3$, then $X$ is an $s$-$m_{1}$-space; (2) If $X$ is a strongly monotonically normal space, then $X$ is an $s$-$m_{2}$-space; (3) If $X$ is a $\sigma$-$m_{3}$ s...

In this paper, some characterizations of pairwise semi-stratifiable spaces are given by means of pairwise g-functions and semi-continuous functions. The pairwise semi-stratifiability of topological ordered C-spaces with semi-stratifiable topology is discussed.

In this paper, we discuss generalized metric properties of paratopological groups. We prove that a paratopological group is sn-metrizable if and only if it is so-metrizable. Moreover, we pose some questions concerning generalized metric properties on paratopological groups.

We show, modifying very slightly the proofs of the recent results of R. Buzyakova [5] about topological groups, that all assertions in [5] are valid in the more general class of rectifiable spaces.

A topological space G is said to be a rectifiable space provided that there are a surjective homeomorphism φ: G × G → G × G and an element e ∈ G such that π 1 {ring operator} φ = π 1 and for every x ∈ G we have φ (x, x)=(x, e), where π 1: G × G → G is the projection to the first coordinate. In this paper, we firstly show that every submaximal recti...

In this paper, we give some characterizations of quasi-pseudo-metrization spaces by means of pairwise weak base g-functions.

We mainly discuss the cardinal invariants and generalized metric properties on paratopological groups or rectifiable spaces, and show that: (1) If $A$ and $B$ are $\omega$-narrow subsets of a paratopological group $G$, then $AB$ is $\omega$-narrow in $G$, which give an affirmative answer for \cite[Open problem 5.1.9]{A2008}; (2) Every bisequential...

When does a topological group $G$ have a Hausdorff compactification $bG$ with a remainder belonging to a given class of spaces? In this paper, we mainly improve some results of A.V. Arhangel'ski\v{\i} and C. Liu's. Let $G$ be a non-locally compact topological group and $bG$ be a compactification of $G$. The following facts are established: (1) If $...

We say that a paratopological group G is pseudobounded (ω-pseudobounded), if for every neighborhood V of the identity element e of G, there exists a natu-ral number n such that G = V n (G = ∞ n=1 V n). In this paper, we mainly dis-cuss the pseudobounded and ω-pseudobounded paratopological groups. First, we give an example to show that a theorem in...

In this paper,\ the authors define a space with an uniform base at non-isolated points, give some characterizations of images of metric spaces by boundary-compact maps, and study certain relationship among spaces with special base properties.\ The main results are the following: (1)\ $X$ is an open,\ boundary-compact image of a metric space if and...

In this paper, the authors mainly discuss the images of spaces with an uniform base at non-isolated points, and obtain the following main results: (1)\ Perfect maps preserve spaces with an uniform base at non-isolated points; (2)\ Open and closed maps preserve regular spaces with an uniform base at non-isolated points; (3)\ Spaces with an uniform b...

In this paper, we mainly introduce the notion of an open uniform (G) at non-isolated points, and show that a space $X$ has an open uniform (G) at non-isolated points if and only if $X$ is the open boundary-compact image of metric spaces. Moreover, we also discuss the inverse image of spaces with an open uniform (G) at non-isolated points. Two quest...

Let $f:X\rightarrow Y$ be a map. $f$ is a {\it sequence-covering map}\cite{Si1} if whenever $\{y_{n}\}$ is a convergent sequence in $Y$ there is a convergent sequence $\{x_{n}\}$ in $X$ with each $x_{n}\in f^{-1}(y_{n})$; $f$ is an {\it 1-sequence-covering map}\cite{Ls2} if for each $y\in Y$ there is $x\in f^{-1}(y)$ such that whenever $\{y_{n}\}$...

In this paper, we define the spaces with a regular base at non-isolated points and discuss some metrization theorems. We firstly show that a space $X$ is a metrizable space, if and only if $X$ is a regular space with a $\sigma$-locally finite base at non-isolated points, if and only if $X$ is a perfect space with a regular base at non-isolated poin...

In this paper, we prove a dichotomy theorem for remainders in compactifications of paratopological groups: every remainder of a paratopological group $G$ is either Lindel\"{o}f and meager or Baire. Moreover, we give a negative answer for a question posed by D. Basile and A. Bella in \cite{B1}, and some questions about remainders of paratopological...

In this paper, we firstly discuss the question: Is $l_{2}^{\infty}$ homeomorphic to a rectifiable space or a paratopological group? And then, we mainly discuss locally compact rectifiable spaces, and show that a locally compact and separable rectifiable space is $\sigma$-compact, which gives an affirmative answer to A.V. Arhangel'ski\v{i} and M.M....

A topological space $G$ is said to be a {\it rectifiable space} provided that there are a surjective homeomorphism $\phi :G\times G\rightarrow G\times G$ and an element $e\in G$ such that $\pi_{1}\circ \phi =\pi_{1}$ and for every $x\in G$ we have $\phi (x, x)=(x, e)$, where $\pi_{1}: G\times G\rightarrow G$ is the projection to the first coordinat...

We mainly introduce some weak versions of the M1M1-spaces, and study some properties about these spaces. The mainly results are that: (1) If X is a compact scattered space and i(X)⩽3i(X)⩽3, then X is an s-m1m1-space; (2) If X is a strongly monotonically normal space, then X is an s-m2m2-space; (3) If X is a σ-m3m3-space, then t(X)⩽c(X)t(X)⩽c(X), wh...

A space X is called a submetrizable space if it can be mapped onto a metric space by a one-to-one map. In this paper, the internal characterizations on certain compact or K-images of submetrizable spaces are discussed. We obtain some characterizations of compact-covering compact images, compact-covering and sequence-covering compact images, sequenc...

In a recent paper [3], D. Buhagiar and B. A. Pasynkov introduced the notion of a supercomplete space and established an internal characterization of these spaces. It is clear that the proof of this characterization actually characterizes c-$$ \mathcal{U} $$-supercomplete spaces. In this short note we state the correct formulation and give a counter...