arXiv subjects

Shou Lin

Publications and source records attributed to Shou Lin.

17 recordsLinked to original sources

arXiv2023

On certain generalized notions using $\mathcal{I}$-convergence in topological spaces

In this paper, we consider certain topological properties along with certain types of mappings on these spaces defined by the notion of ideal convergence. In order to do that, we primarily follow in the footsteps of the earlier studies of ideal convergence done by using functions (from an infinite set $S$ to $X$) in \cite{CS, das4, das5}, as that is the most general perspective and use functions instead of sequences/nets/double sequences etc. This functional approach automatically provides the most general settings for such studies and consequently extends and unifies the proofs of several old and recent results in the literature about spaces like sequential, Fr\'{e}chet-Uryshon spaces and sequential, quotient and covering maps. In particular, we introduce and investigate the notions of $\ic$-functional spaces, $\ic$-functional continuous, quotient and covering mappings and finally $\ic$-functional Fr\'{e}chet-Uryshon spaces. In doing so, we take help of certain set theoretic and other properties of ideals.

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arXiv2022

On $\mathcal{I}$-covering images of metric spaces

Let $\mathcal{I}$ be an ideal on $\mathbb{N}$. A mapping $f:X\to Y$ is called an $\mathcal{I}$-covering mapping provided a sequence $\{y_{n}\}_{n\in\mathbb N}$ is $\mathcal{I}$-converging to a point $y$ in $Y$, there is a sequence $\{x_{n}\}_{n\in\mathbb N}$ converging to a point $x$ in $X$ such that $x\in f^{-1}(y)$ and each $x_n\in f^{-1}(y_n)$. In this paper we study the spaces with certain $\mathcal{I}$-$cs$-networks and investigate the characterization of the images of metric spaces under certain $\mathcal{I}$-covering mappings, which prompts us to discover $\mathcal{I}$-$csf$-networks. The following main results are obtained: (1) A space $X$ has an $\mathcal{I}$-$csf$-network if and only if $X$ is a continuous and $\mathcal{I}$-covering image of a metric space. (2) A space $X$ is an $\mathcal{I}$-$csf$-countable space if and only if $X$ is a continuous $\mathcal{I}$-covering and boundary $s$-image of a metric space. (3) A space $X$ has a point-countable $\mathcal{I}$-$cs$-network if and only if $X$ is a continuous $\mathcal{I}$-covering and $s$-image of a metric space.

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arXiv2017

The $k$-property and countable tightness of free topological vector spaces

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 $f=\overline{f}\circ i$. In this paper the $k$-property and countable tightness of free topological vector space over some generalized metric spaces are studied. The characterization of a space $X$ is given such that the free topological vector space $V(X)$ is a $k$-space or the tightness of $V(X)$ is countable. Furthermore, the characterization of a space $X$ is also provided such that if the fourth level of $V(X)$ has the $k$-property or is of the countable tightness then $V(X)$ is too.

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arXiv2016

The $k_{R}$-property of free Abelian topological groups and products of sequential fans

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 $f$ to each compact subset is continuous. In this paper, we discuss the $k_{R}$-property of products of sequential fans and free Abelian topological groups by applying the $\kappa$-fan introduced by Banakh. In particular, we prove the following two results: (1) The space $S_{\omega_{1}}\times S_{\omega_{1}}$ is not a $k_{R}$-space. (2) The space $S_{\omega}\times S_{\omega_{1}}$ is a $k_{R}$-space if and only if $S_{\omega}\times S_{\omega_{1}}$ is a $k$-space if and only if $\mathfrak b>\omega_1$. These results generalize some well-known results on sequential fans. Furthermore, we generalize some results of Yamada on the free Abelian topological groups by applying the above results. Finally, we pose some open questions about the $k_{R}$-spaces.

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arXiv2016

The $k_{R}$-property on free topological groups

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arXiv2015

Some New Questions on Point-countable Covers and Sequence-covering Mappings

In this survey, 37 questions on point-countable covers and sequence-covering mappings are listed, in which some of these questions have been answered. These questions are mainly related to the theory of generalized metric spaces, involving point-countable covers, sequence-covering mappings, images of metric spaces and hereditarily closure-preserving families.

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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 Lindel\"of $\Sigma$-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 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.

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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 $\omega$-bounded spaces and $r$-spaces), which give some partial answers to a problem posed in [11].

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arXiv2013

$\pi$-metrizable spaces and strongly $\pi$-metrizable spaces

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 $\pi$-metrizable if and only if $X$ is almost $\sigma$-paracompact and locally $\pi$-metrizable; (3) Open and closed maps preserve $\pi$-metrizability; (4) $\pi$-metrizability satisfies hereditarily closure-preserving regular closed sum theorems. Moreover, we define the notions of second-countable $\pi$-metrizable and strongly $\pi$-metrizable spaces, and study some related questions. Some questions about strongly $\pi$-metrizability are posed.

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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 $\sigma$-$m_{3}$ 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.

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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.

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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.

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arXiv2011

Open uniform (G) at non-isolated points and maps

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 questions about open uniform (G) at non-isolated points are posed.

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arXiv2011

Sequence-covering maps on generalized metric spaces

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 $\{y_{n}\}$ is a sequence converging to $y$ in $Y$ there is a sequence $\{x_{n}\}$ converging to $x$ in $X$ with each $x_{n}\in f^{-1}(y_{n})$. In this paper, we mainly discuss the sequence-covering maps on generalized metric spaces, and give an affirmative answer for a question in \cite{LL1} and some related questions, which improve some results in \cite{LL1, Ls4, YP}, respectively. Moreover, we also prove that open and closed maps preserve strongly monotonically monolithity, and closed sequence-covering maps preserve spaces with a $\sigma$-point-discrete $k$-network. Some questions about sequence-covering maps on generalized metric spaces are posed.

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arXiv2011

Regular Bases At Non-isolated Points And Metrization Theorems

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 points, if and only if $X$ is a $\beta$-space with a regular base at non-isolated points. In addition, we also discuss the relations between the spaces with a regular base at non-isolated points and some generalized metrizable spaces. Finally, we give an affirmative answer for a question posed by F. C. Lin and S. Lin in \cite{LL}, which also shows that a space with a regular base at non-isolated points has a point-countable base.

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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 $\sigma$-compact, which gives an affirmative answer to A.V. Arhangel'ski\v{i} 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 $\alpha_{4}$-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}.

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arXiv2011

About remainders in compactifications of paratopological groups

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 groups are posed in the paper.

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