arXiv subjects

Fucai Lin

Publications and source records attributed to Fucai Lin.

At least 55 records · Page 3Linked to original sources

arXiv2016

Countable tightness and $\mathfrak G$-bases on Free topological groups

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 consider two topological properties of $F(X)$ or $A(X)$, namely the countable tightness and $\mathfrak G$-base. We provide some characterizations of the countable tightness and $\mathfrak G$-base of $F(X)$ and $A(X)$ for various special classes of spaces $X$. Furthermore, we also study the countable tightness and $\mathfrak G$-base of some $F_{n}(X)$ of $F(X)$.

math.GN↗

arXiv2016

Two Weak Forms of Countability Axioms in Free Topological Groups

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 basis $X$. In this paper, we discuss two weak forms of countability axioms in $F(X)$ or $A(X)$, namely the $csf$-countability and $snf$-countability. We provide some characterizations of the $csf$-countability and $snf$-countability of $F(X)$ and $A(X)$ for various classes of spaces $X$. In addition, we also study the $csf$-countability and $snf$-countability of $F_n(X)$ or $A_n(X)$, for $n=2, 3, 4$. Some results of Arhangel'ski\vı in \cite{A1980} and Yamada in \cite{Y1998} are generalized. An affirmative answer to an open question posed by Li et al. in \cite{LLL} is provided.

math.GR↗

arXiv2016

The $k$-spaces property of free Abelian topological groups over non-metrizable Lašnev spaces

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 $k$-space for the non-metrizable Lašnev space $X$, which gives a complementary for one result of K. Yamada's. In addition, we also show that, under the assumption of $\flat=ω_1$, the subspace $A_3(X)$ is a $k$-space if and only if $A(X)$ is a $k$-space for the non-metrizable Lašnev space $X$. However, under the assumption of $\flat>ω_1$, we provide a non-metrizable Lašnev space $X$ such that $A_3(X)$ is a $k$-space but $A(X)$ is not a $k$-space.

math.GR↗

arXiv2015

Some topological properties of Charming spaces

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 Lindelöf $Σ$-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 properties of charming spaces and give some characterizations of some class of charming spaces. Finally, we show that the Suslin number of an arbitrary charming rectifiable space $G$ is countable.

math.GN↗

arXiv2015

The topological properties of $q$-spaces in free topological groups

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 $F(X)$, $A(X)$, some finite level of $F(X)$ and some finite level of $A(X)$ is $q$-space (in particular, locally $ω$-bounded spaces and $r$-spaces), which give some partial answers to a problem posed in [11].

math.GR↗

arXiv2015

The mapping $i_{2}$ on the free paratopological groups

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 improve some results of A.S. Elfard and P. Nickolas's [On the topology of free paratopological groups. II, Topology Appl., 160(2013), 220--229.]. The main result is that the natural mapping $i_{2}: (X\bigoplus X_{d}^{-1}\bigoplus\{e\})^{2}\longrightarrow FP_{2}(X)$ is a closed mapping if and only if every neighborhood $U$ of the diagonal $Δ_{1}$ in $X_{d}\times X$ is a member of the finest quasi-uniformity on $X$, where $X$ is a $T_{1}$-space and $X_{d}$ denotes $X$ when equipped with the discrete topology in place of its given topology.

math.GR↗

arXiv2015

Locally $σ$-compact rectifiable spaces

A topological space $G$ is said to be a {\it rectifiable space} provided that there are a surjective homeomorphism $φ:G\times G\rightarrow G\times G$ and an element $e\in G$ such that $π_{1}\circ φ=π_{1}$ and for every $x\in G$, $φ(x, x)=(x, e)$, where $π_{1}: G\times G\rightarrow G$ is the projection to the first coordinate. In this paper, we first prove that each locally compact rectifiable space is paracompact, which gives an affirmative answer to Arhangel'skii and Choban's question (Arhangel'skii and Choban [3]). Then we prove that every locally $σ$-compact rectifiable space with a $bc$-base is locally compact or zero-dimensional, which improves Arhangel'skii and van Mill's result (Arhangel'skii and van Mill [4]). Finally, we prove that each $k_ω$-rectifiable space is rectifiable complete.

math.GN↗

arXiv2013

On paratopological groups

In this paper, we firstly construct a Hausdorff non-submetrizable paratopological group $G$ in which every point is a $G_δ$-set, which gives a negative answer to Arhangel'ski\vı\ and Tkachenko's question [Topological Groups and Related Structures, Atlantis Press and World Sci., 2008]. We prove that each first-countable Abelian paratopological group is submetrizable. Moreover, we discuss developable paratopological groups and construct a non-metrizable, Moore paratopological group. Further, we prove that a regular, countable, locally $k_ω$-paratopological group is a discrete topological group or contains a closed copy of $S_ω$. Finally, we discuss some properties on non-H-closed paratopological groups, and show that Sorgenfrey line is not H-closed, which gives a negative answer to Arhangel'ski\vı\ and Tkachenko's question [Topological Groups and Related Structures, Atlantis Press and World Sci., 2008]. Some questions are posed.

math.GN↗

arXiv2013

$π$-metrizable spaces and strongly $π$-metrizable spaces

A space $X$ is said to be $π$-metrizable if it has a $σ$-discrete $π$-base. In this paper, we mainly give affirmative answers for two questions about $π$-metrizable spaces. The main results are that: (1) A space $X$ is $π$-metrizable if and only if $X$ has a $σ$-hereditarily closure-preserving $π$-base; (2) $X$ is $π$-metrizable if and only if $X$ is almost $σ$-paracompact and locally $π$-metrizable; (3) Open and closed maps preserve $π$-metrizability; (4) $π$-metrizability satisfies hereditarily closure-preserving regular closed sum theorems. Moreover, we define the notions of second-countable $π$-metrizable and strongly $π$-metrizable spaces, and study some related questions. Some questions about strongly $π$-metrizability are posed.

math.GN↗

arXiv2013

A note on free paratopological groups

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.

math.GN↗

arXiv2013

Some weak versions of the $M_{1}$-spaces

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 $σ$-$m_{3}$ space, then $t(X)\leq c(X)$, which extends a result of P.M. Gartside in \cite{CP}. Moreover, some questions are posed in the paper.

math.GN↗

arXiv2013

Topological monomorphisms between 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 $\hat{e}_{X, Y}$ is a topological monomorphism, that is, when $\hat{e}_{X, Y}$ is a topological embedding of $AP(X)$ to $AP(Y)$.

math.GN↗

arXiv2012

Topologically subordered rectifiable spaces and compactifications

A topological space $G$ is said to be a {\it rectifiable space} provided that there are a surjective homeomorphism $ϕ:G\times G\rightarrow G\times G$ and an element $e\in G$ such that $π_{1}\circ ϕ=π_{1}$ and for every $x\in G$ we have $ϕ(x, x)=(x, e)$, where $π_{1}: G\times G\rightarrow G$ is the projection to the first coordinate. In this paper, we mainly discuss the rectifiable spaces which are suborderable, and show that if a rectifiable space is suborderable then it is metrizable or a totally disconnected P-space, which improves a theorem of A.V. Arhangel'ski\vı in \cite{A20092}. As an applications, we discuss the remainders of the Hausdorff compactifications of GO-spaces which are rectifiable, and we mainly concerned with the following statement, and under what condition $Φ$ it is true. Statement: Suppose that $G$ is a non-locally compact GO-space which is rectifiable, and that $Y=bG\setminus G$ has (locally) a property-$Φ$. Then $G$ and $bG$ are separable and metrizable. Moreover, we also consieder some related matters about the remainders of the Hausdorff compactifications of rectifiable spaces.

math.GN↗

arXiv2012

On rectifiable spaces and paratopological groups

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 $ω$-narrow subsets of a paratopological group $G$, then $AB$ is $ω$-narrow in $G$, which give an affirmative answer for \cite[Open problem 5.1.9]{A2008}; (2) Every bisequential or weakly first-countable rectifiable space is metrizable; (3) The properties of Fr$\acute{e}$chet-Urysohn and strongly Fr$\acute{e}$chet-Urysohn are coincide in rectifiable spaces; (4) Every rectifiable space $G$ contains a (closed) copy of $S_ω$ if and only if $G$ has a (closed) copy of $S_{2}$; (5) If a rectifiable space $G$ has a $σ$-point-discrete closed $k$-network, then $G$ contains no closed copy of $S_{ω_{1}}$; (6) If a rectifiable space $G$ is pointwise canonically weakly pseudocompact, then $G$ is a Moscow space. Also, we consider the remainders of paratopological groups or rectifiable spaces, and give a partial answer to questions posed by C. Liu in \cite{Liu2009} and C. Liu, S. Lin in \cite{Liu20091}, respectively.

math.GN↗

arXiv2012

Local properties on the remainders of the topological groups

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\vı 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 $bG\setminus G$ has a locally a point-countable $p$-metabase and $π$-character of $bG\setminus G$ is countable, then $G$ and $bG$ are separable and metrizable; (2) If $bG\setminus G$ has locally a $δθ$-base, then $G$ and $bG$ are separable and metrizable; (3) If $bG\setminus G$ has locally a quasi-$G_δ$-diagonal, then $G$ and $bG$ are separable and metrizable. Finally, we give a partial answer for a question, which was posed by C. Liu in \cite{LC}.

math.GN↗

arXiv2011

A note on rectifiable spaces

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 $σ$-compact, which gives an affirmative answer to A.V. Arhangel'skiǐ and M.M. Choban's question [On remainders of rectifiable spaces, Topology Appl., 157(2010), 789-799]. Next, we show that a rectifiable space $X$ is strongly Fr$\acute{e}$chet-Urysohn if and only if $X$ is an $α_{4}$-sequential space. Moreover, we discuss the metrizabilities of rectifiable spaces, which gives a partial answer for a question posed in \cite{LFC2009}. Finally, we consider the remainders of rectifiable spaces, which improve some results in \cite{A2005, A2007, A2009, Liu2009}.

math.GN↗

arXiv2011

Uniform covers at non-isolated points

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 only if $X$ has an uniform base at non-isolated points; (2)\ Each discretizable space of a space with an uniform base is an open compact and at most boundary-one image of a space with an uniform base; (3)\ $X$ has a point-countable base if and only if $X$ is a bi-quotient,\ at most boundary-one and countable-to-one image of a metric space.

math.GN↗

arXiv2011

Uniform bases at non-isolated points and maps

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 base at non-isolated points don't satisfy the decomposition theorem.

math.GN↗