每个 $T_1$ 连通第一可数空间都是某个连通可度量化空间的连续开映射像

Hyperspaces with a countable character of closed subsets

For a regular space $X$, the hyperspace $(CL(X), τ_{F})$ (resp., $(CL(X), τ_{V})$) is the space of all nonempty closed subsets of $X$ with the Fell topology (resp., Vietoris topology). In this paper, we give the characterization of the space $X$ such that the hyperspace $(CL(X), τ_{F})$ (resp., $(CL(X), τ_{V})$) with a countable character of closed subsets. We mainly prove that $(CL(X), τ_F)$ has a countable character on each closed subset if and only if $X$ is compact metrizable, and $(CL(X), τ_F)$ has a countable character on each compact subset if and only if $X$ is locally compact and sepa…

Sequential rectifiable spaces of countable $cs^*$-character

We prove that each non-metrizable sequential rectifiable space $X$ of countable $cs^*$-character contains a clopen rectifiable submetrizable $k_ω$-subspace $H$ and admits an open disjoint cover by subspaces homeomorphic to clopen subspaces of $H$. This implies that each sequential rectifiable space with countable $cs^*$-character either is metrizable or else is a topological sum of submetrizable $k_ω$-spaces. Consequently, $X$ is submetrizable and paracompact. This answers a question of Lin and Shen posed in 2011.

The convergence-theoretic approach to g-first countable and symmetrizable spaces

This article fits in the context of the approach to topological problems in terms of the underlying convergence space structures, and serves as yet another illustration of the power of the method. More specifically, we spell out convergence-theoretic characterizations of the notions of weak base, weakly first-countable space, semi-metrizable space, and symmetrizable spaces. With the help of the already established similar characterizations of the notions of Fréchet-Ursyohn, sequential, and accessibility spaces, we give a simple algebraic proof of a classical result regarding when a symmetrizab…

Non metrizable topologies on Z with countable dual group

In this paper we give two families of non-metrizable topologies on the group of the integers having a countable dual group which is isomorphic to a infinite torsion subgroup of the unit circle in the complex plane. Both families are related to $D$-sequences, which are sequences of natural numbers such that each term divides the following. The first family consists of locally quasi-convex group topologies. The second consists of complete topologies which are not locally quasi-convex. In order to study the dual groups for both families we need to make numerical considerations of independent inte…

The $C_p$-stable closure of the class of separable metrizable spaces

Denote by $\mathbf C_p[\mathfrak M_0]$ the $C_p$-stable closure of the class $\mathfrak M_0$ of all separable metrizable spaces, i.e., $\mathbf C_p[\mathfrak M_0]$ is the smallest class of topological spaces that contains $\mathfrak M_0$ and is closed under taking subspaces, homeomorphic images, countable topological sums, countable Tychonoff products, and function spaces $C_p(X,Y)$. Using a recent deep result of Chernikov and Shelah (2014), we prove that $\mathbf C_p[\mathfrak M_0]$ coincides with the class of all Tychonoff spaces of cardinality strictly less than $\beth_{ω_1}$. Being motivat…

Countable dense homogeneity and $λ$-sets

We show that all sufficiently nice $λ$-sets are countable dense homogeneous ($\mathsf{CDH}$). From this fact we conclude that for every uncountable cardinal $κ\le \mathfrak{b}$ there is a countable dense homogeneous metric space of size $κ$. Moreover, the existence of a meager in itself countable dense homogeneous metric space of size $κ$ is equivalent to the existence of a $λ$-set of size $κ$. On the other hand, it is consistent with the continuum arbitrarily large that every $\mathsf{CDH}$ metric space has size either $ω_1$ or size $\mathfrak c$. An example of a Baire $\mathsf{CDH}$ metric s…

$π$-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-countabl…

Second-countable compact Hausdorff spaces as remainders in $\mathbf{ZF}$ and two new notions of infiniteness

In the absence of the Axiom of Choice, necessary and sufficient conditions for a locally compact Hausdorff space to have all non-empty second-countable compact Hausdorff spaces as remainders are given in $\mathbf{ZF}$. Among other independence results, the characterization of locally compact Hausdorff spaces having all non-empty metrizable compact spaces as remainders, obtained by Hatzenhuhler and Mattson in $\mathbf{ZFC}$, is proved to be independent of $\mathbf{ZF}$. Urysohn's Metrization Theorem is generalized to the following theorem: every $T_3$-space which admits a base expressible as a …

On the $C_k$-stable closure of the class of (separable) metrizable spaces

Denote by $\mathbf C_k[\mathfrak M]$ the $C_k$-stable closure of the class $\mathfrak M$ of all metrizable spaces, i.e., $\mathbf C_k[\mathfrak M]$ is the smallest class of topological spaces that contains $\mathfrak M$ and is closed under taking subspaces, homeomorphic images, countable topological sums, countable Tychonoff products, and function spaces $C_k(X,Y)$ with Lindelöf domain. We show that the class $\mathbf C_k[\mathfrak M]$ coincides with the class of all topological spaces homeomorphic to subspaces of the function spaces $C_k(X,Y)$ with a separable metrizable space $X$ and a metri…

Metrizable DH-spaces of the first category

We show that if a separable space X has a meager open subset containing a copy of the Cantor set 2^ω, then X has $\frak{c}$ types of countable dense subsets. We suggest a generalization of the λ-set for non-separable spaces. Let X be an h-homogeneous Λ-set. Then X is densely homogeneous and (X \setminus A) is homeomorphic to X for every σ-discrete A \subset X.

Corson 紧致空间幂中的稠密可度量化子空间

Dense metrizable subspaces in powers of Corson compacta

本文刻画了 Corson 紧致空间可数幂存在稠密可度量子空间的条件，构造了反例，并给出了 Kunen 和 van Mill 相关度量化刻画的新证明。

Products and countable dense homogeneity

Building on work of Baldwin and Beaudoin, assuming Martin's Axiom, we construct a zero-dimensional separable metrizable space $X$ such that $X$ is countable dense homogeneous while $X^2$ is not. It follows from results of Hrušák and Zamora Avilés that such a space $X$ cannot be Borel. Furthermore, $X$ can be made homogeneous and completely Baire as well.

Countable dense homogeneity of definable spaces

We investigate which definable separable metric spaces are countable dense homogeneous (CDH). We prove that a Borel CDH space is completely metrizable and give a complete list of zero-dimensional Borel CDH spaces. We also show that for a Borel X subset of 2^omega the following are equivalent: (1) X is G_delta in 2^omega, (2) X^omega is CDH and (3) X^omega is homeomorphic to 2^omega or to omega^omega. Assuming the Axiom of Projective Determinacy the results extend to all projective sets and under the Axiom of Determinacy to all separable metric spaces. In particular, modulo large cardinal assum…

零维可度量化CDH空间$X$使得$X^2$非CDH

Zero-dimensional metrizable CDH space $X$ such that $X^2$ is not CDH

本文利用$λ$-集概念构造出平方空间$X^2$恰有$\frak{c}$个可数稠密子集的零维可度量化CDH空间$X$，且该空间可一致为余解析空间，回答了Medini的开放问题。

Compact complement topologies and k-spaces

Let $(X,τ)$ be a Hausdorff space, where $X$ is an infinite set. The compact complement topology $τ^{\star}$ on $X$ is defined by: $τ^{\star}=\{\emptyset\} \cup \{X\setminus M, \text{where $M$ is compact in $(X,τ)$}\}$. In this paper, properties of the space $(X, τ^{\star})$ are studied in $\mathbf{ZF}$ and applied to a characterization of $k$-spaces, to the Sorgenfrey line, to some statements independent of $\mathbf{ZF}$, as well as to partial topologies that are among Delfs-Knebusch generalized topologies. Among other results, it is proved that the axiom of countable multiple choice (\textbf{…

Mahavier Products, Idempotent Relations, and Condition $Γ$

Clearly, a generalized inverse limit of metrizable spaces indexed by $\mathbb N$ is metrizable, as it is a subspace of a countable product of metrizable spaces. The authors previously showed that all idempotent, upper semi-continuous, surjective, continuum-valued bonding functions on $[0,1]$ (besides the identity) satisfy a certain Condition $Γ$; it follows that only in trivial cases can a generalized inverse limit of copies of ([0,1]) indexed by an uncountable ordinal be metrizable. The authors show that Condition $Γ$ is in fact guaranteed by much weaker criteria, proving a more general metri…

Enumeration degrees and non-metrizable topology

The enumeration degrees of sets of natural numbers can be identified with the degrees of difficulty of enumerating neighborhood bases of points in a universal second-countable $T_0$-space (e.g. the $ω$-power of the Sierpiński space). Hence, every represented second-countable $T_0$-space determines a collection of enumeration degrees. For instance, Cantor space captures the total degrees, and the Hilbert cube captures the continuous degrees by definition. Based on these observations, we utilize general topology (particularly non-metrizable topology) to establish a classification theory of enume…

On strategies for selection games related to countable dimension

Two selection games from the literature, $G_c(\mathcal O,\mathcal O)$ and $G_1(\mathcal O_{zd},\mathcal O)$, are known to characterize countable dimension among certain spaces. This paper studies their perfect- and limited-information strategies, and investigates issues related to non-equivalent characterizations of zero-dimensionality for spaces that are not both separable and metrizable. To relate results on zero-dimensional and finite-dimensional spaces, a generalization of Telgársky's proof that the point-open and finite-open games are equivalent is demonstrated.

On $\mathfrak{P}$-spaces and related concepts

The concept of the strong Pytkeev property, recently introduced by Tsaban and Zdomskyy in [32], was successfully applied to the study of the space $C_c(X)$ of all continuous real-valued functions with the compact-open topology on some classes of topological spaces $X$ including Čech-complete Lindelöf spaces. Being motivated also by several results providing various concepts of networks we introduce the class of $\mathfrak{P}$-spaces strictly included in the class of $\aleph$-spaces. This class of generalized metric spaces is closed under taking subspaces, topological sums and countable product…

Characterizing Subgroups of Compact Abelian Groups

We prove that every countable subgroup of a compact metrizable abelian group has a characterizing set. As an application, we answer several questions on maximally almost periodic (MAP) groups and give a characterization of the class of (necessarily MAP) abelian topological groups whose Bohr topology has countable pseudocharacter.

度量空间上 $C_p$-空间之间的一致连续满射

On uniformly continuous surjections between $C_p$-spaces over metrizable spaces

本文研究了度量空间上 $C_p$-空间之间的一致连续满射，证明了若映射逆有界且定义域具有类维度性质（如零维性等），则值域也具有该性质，从而强化并扩展了已有结论。

Spaces l-Dominated by I or R

If $X$ is compact metrizable and has finite fd-height then the unit interval, $I$, $\ell$-dominates $X$, in other words, there is a continuous linear map of $C_p(I)$ onto $C_p(X)$. If the unit interval $\ell$-dominates a space $X$ then $X$ is compact metrizable and has countable fd-height. Similar results are given for spaces $\ell$-dominated by the reals.

Spaces having a small diagonal

We obtain several results and examples concerning the general question ``When must a space with a small diagonal have a G_delta-diagonal?". In particular, we show (1) every compact metrizably fibered space with a small diagonal is metrizable; (2) there are consistent examples of regular Lindelof (even hereditarily Lindelof) spaces with a small diagonal but no G_delta-diagonal; (3) every first-countable hereditarily Lindelof space with a small diagonal has a G_delta-diagonal; (4) assuming CH, every Lindelof Sigma-space with a small diagonal has a countable network; (5) whether countably compact…

可数扇紧性与群值函数空间中的选择游戏

Countable Fan Tightness and Selection Games in Group-Valued Function Spaces

本文研究了群值函数空间中选择原理与游戏理论之间的等价关系，证明了在特定拓扑群下，某些游戏策略的胜利条件等价，扩展了Clontz的结果并提升了选择原理等价性的理论框架。

Uniform Eberlein compactifications of metrizable spaces

We prove that each metrizable space (of cardinality less or equal to continuum) has a (first countable) uniform Eberlein compactification and each scattered metrizable space has a scattered hereditarily paracompact compactification. Each compact scattered hereditarily paracompact space is uniform Eberlein and belongs to the smallest class of compact spaces, that contain the empty set, the singleton, and is closed under producing the Aleksandrov compactification of the topological sum of a family of compacta from that class.

拟连续函数空间的完备性性质

Completeness properties of the space of quasicontinuous functions

本文研究配备逐点收敛拓扑的拟连续函数空间的完备性性质，明确其完全可度量化、波兰性与Čech完备性等价，并给出该空间完全可度量化的充要条件。

两个非平凡可数维连续统的乘积不能轻映射到任何因子中

No product of two non-trivial countable-dimensional continua maps lightly into any of the factors

本文证明了正维数紧可度量化空间的乘积在特定连续映射下，其因子必包含无一维子集的非平凡连续统，从而对 J. Dudák 和 B. Vejnar 的问题给出了否定回答。

Topological properties of some function spaces

Let $Y$ be a metrizable space containing at least two points, and let $X$ be a $Y_{\mathcal{I}}$-Tychonoff space for some ideal $\mathcal{I}$ of compact sets of $X$. Denote by $C_{\mathcal{I}}(X,Y)$ the space of continuous functions from $X$ to $Y$ endowed with the $\mathcal{I}$-open topology. We prove that $C_{\mathcal{I}}(X,Y)$ is Fréchet - Urysohn iff $X$ has the property $γ_{\mathcal{I}}$. We characterize zero - dimensional Tychonoff spaces $X$ for which the space $C_{\mathcal{I}}(X,{\bf 2})$ is sequential. Extending the classical theorems of Gerlits, Nagy and Pytkeev we show that if $Y$ i…

$\mathfrak P_0$-spaces

A regular topological space $X$ is defined to be a $\mathfrak P_0$-space if it has countable Pytkeev network. A network $\mathcal N$ for $X$ is called a Pytkeev network if for any point $x\in X$, neighborhood $O_x\subset X$ of $x$ and subset $A\subset X$ accumulating at a $x$ there is a set $N\in\mathcal N$ such that $N\subset O_x$ and $N\cap A$ is infinite. The class of $\mathfrak P_0$-spaces contains all metrizable separable spaces and is (properly) contained in the Michael's class of $\aleph_0$-spaces. It is closed under many topological operations: taking subspaces, countable Tychonoff pro…

Metrizability of CHART groups

For compact Hausdorff admissible right topological (CHART) group $G$, we prove $w(G)=πχ(G)$. This equality is well known for compact topological groups. This implies the criteria for the metrizability of CHART groups: if $G$ is first-countable (2013, Moors, Namioka) or $G$ is Fréchet (2013, Glasner, Megrelishvili), or $G$ has countable $π$-character (2022, Reznichenko) then $G$ is metrizable. Under the continuum hypothesis (CH) assumption, a sequentially compact CHART group is metrizable. Namioka's theorem that metrizable CHART groups are topological groups extends to CHART groups with small w…

Locally $n$-connected compacta and $UV^n$-maps

We provide a machinery for transferring some properties of metrizable $ANR$-spaces to metrizable $LC^n$-spaces. As a result, we show that for complete metrizable spaces the properties $ALC^n$, $LC^n$ and $WLC^n$ coincide to each other. We also provide the following spectral characterizations of $ALC^n$ and cell-like compacta: A compactum $X$ is $ALC^n$ if and only if $X$ is the limit space of a $σ$-complete inverse system $S=\{X_α, p^β_α, α<β<τ\}$ consisting of compact metrizable $LC^n$-spaces $X_α$ such that all bonding projections $p^β_α$, as a well all limit projections $p_α$, are $UV^n$-ma…

Some generalized metric properties on hyperspaces with the Vietoris topology

We study the heredity of the classes of generalized metric spaces (for example, spaces with a $σ$-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, $γ$-spaces, semi-metrizable spaces) to the hyperspaces of nonempty compact subsets and finite subsets with the Vietoris topology.

Universal metric spaces and extension dimension

For any countable $CW$-complex $K$ and a cardinal number $τ\geqω$ we construct a completely metrizable space $X(K,τ)$ of weight $τ$ with the following properties: $\e X(K,τ)\leq K$, $X(K,τ)$ is an absolute extensor for all normal spaces $Y$ with $\e Y\leq K$, and for any completely metrizable space $Z$ of weight $\leqτ$ and $\e Z\leq K$ the set of closed embeddings $Z\to X(K,τ)$ is dense in the space $C(Z,X(K,τ))$ of all continuous maps from $Z$ into $X(K,τ)$ endowed with the limitation topology. This result is applied to prove the existence of universal spaces for all metrizable spaces of giv…

拓扑 gyrogroup 商空间上的弱可数性公理

Weak countability axioms on the quotient spaces of topological gyrogroups

研究强拓扑 gyrogroup 商空间的弱可数性公理，在闭强子 gyrogroup 且中性的条件下刻画其双径向性、嵌套性与可度量化关系。

Topological classification of zero-dimensional $M_ω$-groups

A topological group $G$ is called an $M_ω$-group if it admits a countable cover $\K$ by closed metrizable subspaces of $G$ such that a subset $U$ of $G$ is open in $G$ if and only if $U\cap K$ is open in $K$ for every $K\in\K$. It is shown that any two non-metrizable uncountable separable zero-dimenisional $M_ω$-groups are homeomorphic. Together with Zelenyuk's classification of countable $k_ω$-groups this implies that the topology of a non-metrizable zero-dimensional $M_ω$-group $G$ is completely determined by its density and the compact scatteredness rank $r(G)$ which, by definition, is equa…

The topological structure of (homogeneous) spaces and groups with countable cs*-character

In this paper we introduce and study three new cardinal topological invariants called the cs*, cs-, and sb-characters. The class of topological spaces with countable cs*-character is closed under many topological operations and contains all aleph-spaces and all spaces with point-countable cs*-network. Our principal result states that each non-metrizable sequential topological group with countable cs*-character has countable pseudo-character and contains an open $k_ω$-subgroup.

拟连续函数空间的可数紧性类型性质

On countable tightness type properties of spaces of quasicontinuous functions

本文利用选择原理刻画拟连续函数空间的多种可数紧性，并给出κ+-Lusin空间的函数刻画。

The strong Pytkeev property and strong countable completeness in (strongly) topological gyrogroups

A topological gyrogroup is a gyrogroup endowed with a topology such that the binary operation is jointly continuous and the inverse mapping is also continuous. In this paper, it is proved that if $G$ is a sequential topological gyrogroup with an $ω^ω$-base, then $G$ has the strong Pytkeev property. Moreover, some equivalent conditions about $ω^ω$-base and strong Pytkeev property are given in Baire topological gyrogroups. Finally, it is shown that if $G$ is a strongly countably complete strongly topological gyrogroup, then $G$ contains a closed, countably compact, admissible subgyrogroup $P$ su…

A class of quotient spaces in strongly topological gyrogroups

Quotient space is a class of the most important topological spaces in the research of topology. In this paper, we show that if G is a strongly topological gyrogroup with a symmetric neighborhood base U at 0 and H is an admissible subgyrogroup generated from U , then G/H is first-countable if and only if it is metrizable. Moreover, if H is neutral and G/H is Frechet-Urysohn with an ωω-base, then G/H is first-countable. Therefore, we obtain that if H is neutral, then G/H is metrizable if and only if G/H is Frechet-Urysohn with an ωω-base. Finally, it is shown that if H is neutral, π\c{hi}(G/H) =…

Frechet-Urysohn property of quasicontinuous functions

The aim of this paper is to study the Frechet-Urysohn property of the space $Q_p(X,\mathbb{R})$ of real-valued quasicontinuous functions, defined on a Hausdorff space $X$, endowed with the pointwise convergence topology. It is proved that under Suslin's Hypothesis, for an open Whyburn space $X$, the space $Q_p(X,\mathbb{R})$ is Frechet-Urysohn if and only if $X$ is countable. In particular, it is true in the class of first-countable regular spaces $X$. In ZFC, it is proved that for a metrizable space $X$, the space $Q_p(X,\mathbb{R})$ is Frechet-Urysohn if and only if $X$ is countable.

The complexity of homeomorphism relations on some classes of compacta

We prove that the homeomorphism relation between compact spaces can be continuously reduced to the homeomorphism equivalence relation between absolute retracts which strengthens and simplifies recent results of Chang and Gao, and Cieśla. It follows then that the homeomorphism relation of absolute retracts is Borel bireducible with the universal orbit equivalence relation. We also prove that the homeomorphism relation between regular continua is classifiable by countable structures and hence it is Borel bireducible with the universal orbit equivalence relation of the permutation group on a coun…

Covering versus partitioning with the Cantor space

What topological spaces can be partitioned into copies of the Cantor space $2^ω$? An obvious necessary condition is that a space can be partitioned into copies of $2^ω$ only if it can be covered with copies of $2^ω$. We prove three theorems concerning when this necessary condition is also sufficient. If $X$ is a metrizable space and $|X| \leq \mathfrak{c}^{+ω}$ (the least limit cardinal $>\!\mathfrak{c}$), then $X$ can be partitioned into copies of $2^ω$ if and only if $X$ can be covered with copies of $2^ω$. To show this cardinality bound is sharp, we construct a metrizable space of size $\ma…

乘积空间中相对函数可数子集

Relatively functionally countable subsets of products

本文证明在满足每个因子空间的所有RFC子集可数的条件下，Tychonoff乘积空间中所有RFC子集可数，并利用构造连续单射的工具得出可度量空间中RFC子集可数。

Compactifications and universal spaces in extension theory

We show that for each countable simplicial complex P the following conditions are equivalent: (1) $P \in AE(X)$ iff $P \in AE(βX)$ for any space X; (2) There exists a P-invertible map of a metrizable compactum X with $P \in AE(X)$ onto the Hilbert cube.

A Note on Monotonically Metacompact Spaces

We show that any metacompact Moore space is monotonically metacompact and use that result to characterize monotone metacompactness in certain generalized ordered (GO)spaces. We show, for example, that a generalized ordered space with a sigma-closed-discrete dense subset is metrizable if and only if it is monotonically (countably) metacompact, that a monotonically (countably) metacompact GO-space is hereditarily paracompact, and that a locally countably compact GO-space is metrizable if and only if it is monotonically (countably) metacompact. We give an example of a non-metrizable LOTS that is …

Connected economically metrizable spaces

A topological space is nonseparably connected if it is connected but all of its connected separable subspaces are singletons. We show that each connected sequential topological space X is the image of a nonseparably connected complete metric space Eco(X) under a monotone quotient map. The metric d of the space Eco(X) is economical in the sense that for each infinite subspace A of X the cardinality of the set {d(a,b):a,b in A} does not exceed the density of A. The construction of the space Eco(X) determines a functor Eco from the category Top of topological spaces and their continuous maps into…

可数稠密齐次拓扑向量空间是 Baire 空间

A countable dense homogeneous topological vector space is a Baire space

本文证明含 Cantor 集副本的齐次可数稠密齐次空间为 Baire 空间，并据此判定若干函数空间与弱拓扑 Banach 空间不具可数稠密齐次性。

The connected countable spaces of Bing and Ritter are topologically homogeneous

Answering a problem posed by the second author on Mathoverflow, we prove that the connected countable Hausdorff spaces constructed by Bing and Ritter are topologically homogeneous.

Suprema of continuous functions on connected spaces

Let $K$ be a compact Hausdorff space and let $(f_n)_{n\in \N}$ be a pairwise disjoint sequence of continuous functions from $K$ into $[0,1]$. We say that a compact space $L$ \emph{adds supremum} of $(f_n)_{n\in \N}$ in $K$ if there exists a continuous surjection $π:L\longrightarrow K$ such that there exists $sup\{f_n\circπ:n\in \N\}$ in $C(L)$. Moreover, we expect that $L$ preserves suprema of disjoint continuous functions which already existed in $C(K)$. Namely, if $sup\{g_n:n\in \N\}$ exists in $C(K)$, we must have $sup\{g_n\circπ:n\in \N\}$ in $C(L)$. This paper studies the preservation of …

Equilateral Sets in Banach Spaces of th form C(K)

We show that for "most" compact non metrizable spaces, the unit ball of the Banach space C(K) contains an uncountable 2-equilateral set. We also give examples of compact non metrizable spaces K such that the minimum cardinality of a maximal equilateral set in C(K) is countable.

Quotient spaces with strong subgyrogroups

In this paper, we mainly investigate the quotient spaces G/H when G is a strongly topological gyrogroup and H is a strong subgyrogroup of G. It is shown that if G is a strongly topological gyrogroup, H is a closed strong subgyrogroup of G and H is inner neutral, then the quotient space G/H is first-countable if and only if G/H is a bisequential space if and only if G/H is a weakly first-countable space if and only if G/H is a csf-countable and sequential a7-space. Moreover, it is shown that if H is a locally compact metrizable strong subgyrogroup of G and the quotient space G/H is sequential, …

The topological type of spaces consisting of certain metrics on locally compact metrizable spaces with the compact-open topology

For a separable locally compact but not compact metrizable space $X$, let $αX = X \cup \{x_\infty\}$ be the one-point compactification with the point at infinity $x_\infty$. We denote by $EM(X)$ the space consisting of admissible metrics on $X$, which can be extended to an admissible metric on $αX$, endowed with the compact-open topology. Let $ \mathbf{c}_0 \subset (0,1)^\mathbb{N}$ be the space of sequences converging to $0$. In this paper, we shall show that if $X$ is separable, locally connected and locally compact but not compact, and there exists a sequence $\{C_i\}$ of connected sets in …

On a question of Dikranjan and Zava

Let $G$ be a non-discrete countable metrizable abelian topological group endowed with the coarse structure $ \mathcal{C} $ generated by compact subsets of $G$. We prove that $asdim (G, \mathcal{C} ) = \infty$. For an infinite cyclic subgroup $G$ of the circle, this answers a question of Dikranjan and Zava [3].

Extension of continuous functions and weakly developable spaces

The aim of this paper is to investigate weakly developable spaces. For a comparison with semi-metrizable spaces, we introduce and study a class of spaces among those of weakly developable spaces, semimetrizable spaces and first countable spaces having a G_delta-diagonal. Some results are obtained and applications to the problem of extending continuous functions are discussed.