Chuan Liu

Ohio University · Department of Mathematics

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...

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...

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...

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...

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...

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.].

Based on the notions of T. Banakh's strict Pytkeev networks and A.V. Arhangel'skiı̌'s sensor families, strict Pytkeev networks with sensors are introduced in this paper. A family P of subsets of a topological space X is called a strict Pytkeev network with sensors (abbr. an sp-network) if, for each x∈U∩A‾ with U open and A subset in X, there is a 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...

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...

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...

Let F(X) denote the free topological group over a Tychonoff space X, F-n, (X) denote the subspace of F(X) that consists of all words of reduced length <= n with respect to the free basis X for every non-negative integer n and E-n(X) = F-n(X)\Fn-1(X) for n >= 1. In this paper, we study topological properties of free topological groups in terms of Ar...

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...

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].

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...

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, 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 investigate copies of SωSω and S2S2 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 F5(X)F5(X) is Fréchet–Urysohn, which generalizes Yamada's theorem (Yamada [26]). We also give a negative answer to Yamada's conjecture (Yama...

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...

We investigate some generalized metric space properties on paratopological (semitopological) groups and prove that a paratopological group that is quasi-metrizable by a left continuous, left-invariant quasi-metric is a topological group and give a negative answer to Ravskyʼs question (Ravsky, 2001 [18, Question 3.1]). It is also shown that an uncou...

In this paper, we characterize an ℵ 0 -weakly first-countable space as a quotient, countable-to-one image of a first-countable space. We also discuss metrizability and mapping theorems on ℵ 0 -weak bases, and pose some questions.

In terms of General Topology, we consider ordered additive groups having the order topology, including ordered fields. Namely, we investigate metrizability of these groups or fields, and topological properties of ordered fields in terms of Archimedes' axiom or the axiom of continuity. Also, we give a negative answer to a question in [9]. Finally, w...

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....

We discuss generalized metrizable properties on paratopological groups and topological groups. It is proved in this paper that a first-countable paratopological group which is a β-space is developable; and we construct a Hausdorff, separable, non-metrizable paratopological group which is developable. We consider paratopological (topological) groups...

In this paper, we consider the following question: when does a topological group G have a Hausdorff compactification bG with a remainder belonging to a given class of spaces? We extend the results of A.V. Arhangel'skii by showing that if a remainder of a non-locally compact topological group G has a countable open point-network or a locally GδGδ-di...

Let / : X - Y be a closed mapping, where X is a fc-semistratifiable fc-space. If Y contains no closed copy of Su,1 (resp. S), then df∼1(y) is Lindel6f(resp. compact) for each y 6E Y. This improves some results about closed mappings on generalized metric spaces obtained by Liu [10], Tanaka [13, 14, 15], Tanaka and Liu [16], and Yun [19]. At last, tw...

In this paper, it is proved that a space with a point-countable base is an open, countable-to-one image of a metric space, and a quotient, countable-to-one image of a metric space is characterized by a point-countable ℵ0-weak base. Examples are provided in order to answer negatively questions posed by Gruenhage et al. [G. Gruenhage, E. Michael, Y....

We discuss images of metric spaces by quotient mappings such that the cardinality of the boundary of each fiber is at most one and prove that for a space X with a point-countable weak base, there are a metric space M and a quotient s-map f:M→X with |∂f -1 (x)|≤1 for x∈X, which gives an affirmative answer to a question raised by Solin.

In this paper, we characterize closed σ-compact images of metric spaces as Fréchet, weakly quasi-first-countable and N-spaces; and discuss the fibers of closed mappings on some generalized metric spaces and improve some Tanaka's results [22, 24] and answered a question in Yun [28]. It is also shown that g-metrizable spaces and spaces with a point-c...

This chapter presents some results on various generalizations of metric spaces and their quotient spaces in terms of k-networks, weak bases, and weak topologies. The chapter poses some related problems. A collection p{hook} of subsets of a space X is a network (or net) for X if, whenever x ∈U with U open, there is P ∈p{hook} with x ∈P ⊂U. A space i...

In this paper some funcions of product and mapping properties of k-spaces with a point-countable k-network are discussed via special metric spaces T_\omega and T_{\omega_1}. The relationalships among the three classes of spaces, which are introduced respectively for solving the "the products of k-spaces question" posed by Michael in 1973, are analy...

In this paper, we discuss the countable tightness of products of spaces which are quotient s–images of locally separable metric spaces, or k–spaces with a star–countable k–network. The main result is that the following conditions are equivalent: (1) b = ω 1; (2) t(S ω × S ω1 ) > ω; (3) For any pair (X, Y ), which are k–spaces with a point–countable...

In this paper, we give an affirmative answer to Tanaka's question: Is a space X with a σ-hereditarily closure-preserving weak base g-metrizable? [Proc. Aroc. Amer. Math. Soc. 112 (1991) 283] and a negative answer to S. Lin's question: Is every weak base of a topological space a k-network? [S. Lin, Generalized Metric Spaces and Maps, Science Press,...

(Arhangel'skii, 1999) A Hausdorff topological space X is called κ-Fréchet Urysohn if for every open subset A of X and every x ∈ Ā there exists a sequence of points of A converging to x. We discuss the properties of κ-Fréchet Urysohn spaces, the conditions under which a κ-Fréchet Urysohn space is Frédict Urysohn, and the behavior of κ-Fréchet Urysoh...

We give an example of a separable space with a point-countable weak base that is not g-second countable. Also, we give some partial answers to the problem: Is a space with σ-hereditarily closure preserving weak bases g-metrizable?

As a generalization of developments of (developable) spaces, we introduce the notion of σ-strong networks of spaces, and related notions. We give new characterizations for around quotient compact images of metric spaces by means of σ-strong networks, and give some related matters.

Shibakov proved in 1998 that every sequential topological group with a point-countable k-network is metrizable if its sequential order is less than ω1. In this paper, we study further topological properties of sequential topological groups with a point-countable k-network. We introduce the notion of a cs-cover, and show that: If G is a sequential t...

Weakly bisequential spaces were introduced by A.V. Arhangel'skii (1), in this paper. We discuss the relations between weakly bisequential spaces and metric spaces, countably bisequential spaces, Frechet-Urysohn spaces.

In the theory of generalized metric spaces, the notion of k- networks has played an important role. Every locally separable metric space or CW-complex, more generally, every space dominated by locally separa- ble metric spaces has a star-countable k-network. Every LaSnev space, as well as, every space dominated by LaSnev spaces has a a-compact-fini...

This paper is devoted to Aull’s D 1 spaces, i.e., spaces in which every closed subset has a countable local base. The characterization, covering properties and the metrization of D 1 spaces are considered. Some open questions are raised at the end of the paper.

In this paper, we show that a k-space with σ-hereditarily closure preserving k-network or a k- and ℵ-space is a g-metrizable space iff it contains no closed copy of Sω. If answers a question of Lin in 1990. Using this result, we prove that g-metrizable spaces are preserved by open and closed mappings.

We prove the following two results: (1) A T 2 topological space has a locally countable network if and only if it is an ss-image of a metric space; (2) A T 3 topological space has a locally countable weak base if and only if it is a quotient, compact (or quotient, π-) ss-image of a metric space.

Thesis (Ph. D.)--Ohio University, 2002.