arXiv subjects
Publications and source records attributed to Li-Hong Xie.
In this paper, we prove that for every topological MV-algebra the topology is induced by a uniformity. As a application, we obtain that every topological MV-algebra is completely regular and every \(T_0\) first countable topological MV-algebra is metrizable.
We prove that the indicator functions form a uniformly continuous retract of the set of $[0,1]$-valued functions in $L^2([0,1])$. Consequently, the Lebesgue probability measure algebra, equipped with the symmetric-difference metric, is an injective uniform space. The construction uses dyadic conditional expectations and a common exponential coupling to obtain Lipschitz maps into the indicator functions that approximate the identity uniformly on that set. As an application, we construct a Polish topological group whose left uniform space is injective but which admits a bounded left uniformly continuous real-valued function that is not right uniformly continuous. This answers the injective part of a question of Bouziad and Troallic in the negative. The group is a semidirect product of two copies of the measure algebra and is also bounded, locally finite, and of exponent at most four.
We construct a Hausdorff quasitopological abelian group that is second-countable and semiregular, with a bounded continuous function that does not factor through any regular first-countable semitopological group. Its regular and Tychonoff reflections coincide, have a countable network, and fail even factorizability through arbitrary first-countable semitopological groups. This gives negative answers to \cite[Problems~6.1, 6.2, 6.4, and~6.5]{Tk2026} of Tkachenko. The construction starts with a countable cone group whose common separation reflection is not Fréchet--Urysohn, answering \cite[Problems~6.8 and~6.9]{Tk2026} negatively. In contrast, symmetrization gives affirmative answers to \cite[Problems~6.6 and~6.7]{Tk2026} without separation assumptions. We also show that every countable semitopological group admits simultaneous factorizations of countable families through second-countable Tychonoff semitopological groups.
We address three problems on separable quotients of topological groups posed by Leiderman, Morris, and Tkachenko in \cite{LMT} published on Israel Journal of Mathematics. First, we construct in ZFC a connected Baire Pontryagin-reflexive dense subgroup of $\T^{\cc}$ whose countable subgroups are $h$-embedded and whose uncountable subgroups are dense. Its underlying abstract group is the circle group, and all its compact subsets are finite. Second, we construct a zero-dimensional Baire Pontryagin-reflexive example with the same subgroup properties whose underlying group is free abelian of rank $\cc$. Both examples have no nontrivial separable Hausdorff quotient. Third, for the group constructed in their Theorem~3.5, we determine every closed subgroup of every finite power up to an integral change of coordinates and prove that every countable subgroup of every Hausdorff quotient of a finite power is $h$-embedded and closed. The same conclusions hold for our free Baire reflexive example. These results answer Problem~1.25 negatively, Problem~3.12 affirmatively and realize all three regularity properties in Problem~3.14 simultaneously in \cite{LMT}.
A topological group is functionally balanced if every bounded real-valued left uniformly continuous function is right uniformly continuous. We prove that every Hausdorff functionally balanced group has coinciding left and right uniformities, giving an affirmative answer to the Itzkowitz problem. Consequently, the classes of SIN, SFSIN and FSIN groups coincide.
We prove that every countable left uniformly discrete subset of a Hausdorff functionally balanced topological group is right thin. As applications, we answer two questions of Bouziad and Troallic: every left precompact subset of a Hausdorff functionally balanced group is right precompact, and every Hausdorff $ω$-narrow functionally balanced group is a SIN group.
Recently, topological MV-algebras have been investigated by several mathematicians. In this paper, we mainly show that for every Hausdorff topological MV-algebra $A$, there exists a natural topological isomorphism $i_A:A\rightarrow A^\bullet$ of $A$ onto a closed subalgebra of the pathwise connected, locally pathwise connected topological MV-algebra $A^\bullet$. Furthermore, we show that there is an extension to a bounded continuous function on the MV-algebra $A^\bullet$ for each continuous real-valued bounded function on a topological MV-algebra $A$. Finally, we prove that if $φ:A_1\rightarrow A_2$ is a continuous homomorphism of topological MV-algebras, then $φ$ admits a natural extension to a continuous homomorphism $φ^\bullet:A_1^\bullet\rightarrow A_2^\bullet$; in addition, if $φ$ is open and onto, then so is $φ^\bullet$.
Recently, topological MV-algebras have been investigated by several mathematicians. In this paper, we find that every topological \(MV\)-algebra is a Mal'tsev space introduced by Mal'tsev in 1954. Hence, applying the theorem of Reznichenko and Uspenskij on pseudocompact Mal'tsev spaces, we show that the product of arbitrary family of pseudocompact topological \(MV\)-algebras are pseudocompact. We also prove that every $σ$-compact topological \(MV\)-algebra is ccc. Secondly, we obtain that the Stone-Čech compactification of a pseudocompact topological \(MV\)-algebra carries a natural compact topological \(MV\)-algebra structure extending the original one. Finally, we prove that: let \(I\) be a closed ideal in a pseudocompact topological \(MV\)-algebra \(A\) and \(ι_1:A\hookrightarrow\bet A\) is the naturally injective; then \(\cl_{βA}ι_1(I)\) is a closed ideal of \(βA\) and \( βA/\cl_{βA}ι_1(I)\cong β(A/I)\).
For a topological group, the quotient map modulo a subgroup is open and the quotient map modulo a compact subgroup is perfect. In this paper we prove and develop the corresponding compact-ideal theory for topological \(MV\)-algebras. We show that if \(I\) is an ideal of a topological \(MV\)-algebra \(A\), then the natural quotient homomorphism \(q:A\longrightarrow A/I\), where \(A/I\) is endowed with the quotient topology, is a continuous open quotient map and \(A/I\) is again a topological \(MV\)-algebra. If, in addition, \(I\) is compact, then \(q\) is perfect. As applications, we study three-space phenomena in topological \(MV\)-algebras. Under compact-kernel hypotheses we prove three-space theorems for compactness, local compactness, \(σ\)-compactness, Lindelöfness and paracompactness under the separation hypotheses stated below. We also prove a first-countability three-space theorem for locally convex topological \(MV\)-algebras.
In this paper, we mainly prove that if $H$ is a closed strong subgyrogroup of a strongly topological gyrogroup $G$ and $H$ is neutral, then (1) $G/H$ is biradial if and only if $G/H$ is nested; (2) $G/H$ is metrizable if and only if $G/H$ is a biradial space with countable pseudocharacter; (3) $G/H$ is metrizable if and only if $G/H$ has countable $cn$-character, given that $G/H$ has the Baire property.
We present a characterization of paratopological gyrogroups that can be topologically embedded as subgyrogroups into a product of first-countable $T_{i}$ paratopological gyrogroups for $i = 0, 1, 2$. Specifically, we demonstrate that a strongly paratopological gyrogroup $G$ is topologically isomorphic to a subgyrogroup of a topological product of first-countable $T_1$ strongly paratopological gyrogroups if and only if $G$ is $T_1$, $ω$-balanced and the weakly Hausdorff number of $G$ is countable. This means that for every neighborhood $U$ of the identity 0 in $G$, there exists a countable family $γ$ of neighborhoods of 0 such that for all $V \inγ$, $\bigcap_{V\inγ} (\ominus V)\subseteq U$. Similarly, we prove that a strongly paratopological gyrogroup $G$ is topologically isomorphic to a subgyrogroup of a topological product of first-countable Hausdorff strongly paratopological gyrogroups if and only if $G$ is Hausdorff, $ω$-balanced and the Hausdorff number of $G$ is countable. This means that for every neighborhood $U$ of the identity 0 in $G$, there exists a countable family $γ$ of neighborhoods of 0 such that for all $V \inγ$, $\bigcap_{V\inγ} (V\boxminus V)\subseteq U$.
In this paper, we study the uniformities on the double coset spaces in topological groups. As an implication, the quotient spaces of topological groups with a $q$-point are studied. It mainly shows that: (1) Suppose that $G$ is a topological group with a $q$-point and $H$ is a closed subgroup of $G$; then the quotient space $G/H$ is an open and quasi-perfect preimage of a metrizable space; in particular, $G/H$ is an $M$-space. (2) Suppose that $G$ is a topological group with a strict $q$-point and $H$ is a closed subgroup of $G$; then the quotient space $G/H$ is an open and sequentially perfect preimage of a metrizable space. (3) Suppose that $G$ is a topological group with a strong $q$-point and $H$ is a closed subgroup of $G$; then the quotient space $G/H$ is an open and strongly sequentially perfect preimage of a metrizable space.
In this paper, we study some characterizations of $q$-spaces, strict $q$-spaces and strong $q$-spaces under $ω$-balanced topological groups as follows: (1) A topological group $G$ is $ω$-balanced and a $q$-space if and only if for each open neighborhood $O$ of the identity in $G$, there is a countably compact invariant subgroup $H$ which is of countable character in $G$, such that $H \subseteq O$ and the canonical quotient mapping $p:G\rightarrow G/H$ is quasi-perfect and the quotient group $G/H$ is metrizable. (2) A topological group $G$ is $ω$-balanced and a strict $q$-space if and only if for each open neighborhood $O$ of the identity in $G$, there is a closed sequentially compact invariant subgroup $H$ which is of countable character in $G$, such that $H \subseteq O$ and the canonical quotient mapping $p:G\rightarrow G/H$ is sequential-perfect and the quotient group $G/H$ is metrizable. (3) A topological group $G$ is $ω$-balanced and a strong $q$-space if and only if for each open neighborhood $O$ of the identity in $G$, there is a closed sequentially compact invariant subgroup $H$ of countable character $\{V_{n}:n\in ω\} $, such that $H \subseteq O$ and $\{V_{n}:n\inω\}$ is a strong $q$-sequence at each $ y\in H $, in $G$ such that the canonical quotient mapping $p:G\rightarrow G/H$ is strongly sequential-perfect and the quotient group $G/H$ is metrizable.
In this paper, we consider the continuity of the inverse in (strongly) paratopological gyrogroups. The conclusions are established as follows: (1) A compact Hausdorff paratopological gyrogroup $G$ is a topological gyrogroup. (2) A Hausdorff locally compact strongly paratopological gyrogroup is a topological gyrogroup. (3) If $G$ is locally compact strongly paratopological gyrocommutative gyrogroup (without any separation restrictions), then $G$ is a strongly topological gyrogroup. (4) Every regular feebly compact strongly paratopological gyrogroup is a topological gyrogroup. (5) If a Hausdorff strongly paratopological gyrogroup $G$ is countablly compact and topologically periodic, then $G$ is a strongly topological gyrogroup.
A topological gyrogroup is a gyrogroup endowed with a compatible topology such that the multiplication is jointly continuous and the inverse is continuous. In this paper, we study the quotient gyrogroups in topological gyrogroups with respect to strongly $L$-subgyrogroups, and prove that let $(G, τ,\oplus)$ be a topological gyrogroup and $H$ a closed strongly $L$-subgyrogroup of $G$, then the natural homomorphism $π$ from a topological gyrogroup $G$ to its quotient topology on $G/H$ is an open and continuous mapping, and $G/H$ is a homogeneous $T_1$-space. We also establish that for a locally compact strongly $L$-subgyrogroup $H$ of a topological gyrogroup $G$, the natural quotient mapping $π$ of $G$ onto the quotient space $G/H$ is a locally perfect mapping. This leads us to some interesting results on how properties of $G$ depend on the properties of $G/H$. Some classical results in topological groups are generalized.
Metric spaces are generalized by many scholars. Recently, Khatami and Mirzavaziri use a mapping called $t$-definer to popularize the triangle inequality and give a generalization of the notion of a metric, which is called a $\star$-metric. In this paper, we prove that every $\star$-metric space is metrizable. Also, we study the total boundedness and completeness of $\star$-metric spaces.
The concept of gyrogroups is a generalization of groups which do not explicitly have associativity. Recently, Atiponrat extended the idea of topological (paratopological) groups to topological (paratopological) gyrogroups. In this paper, we prove that every regular (Hausdorff) locally gyroscopic invariant paratopological gyrogroup $G$ is completely regular (function Hausdorff). These results improve theorems of Banakh and Ravsky for paratopological groups. Also, we extend the Pontrjagin conditions of (para)topological groups to (para)topological gyrogroups.
The concept of gyrogroups is a generalization of groups which do not explicitly have associativity. Recently, Wattanapan et al consider the construction of Hartman-Mycielski in strongly topological gyrogroups. In this paper, we extend their results in topological gyrogroups. We mainly, among other results, prove that every Hausdorff topological gyrogroup $G$ can be embedded as a closed subgyrogroup of a Hausdorff path-connected and locally path-connected topological gyrogroup $G^\bullet$.