Gary Gruenhage, J. Tatch Moore
A space X is called an alpha-Toronto space if X is scattered of Cantor-Bendixson rank alpha and is homeomorphic to each of its subspaces of same rank. We answer a question of Steprans by constructing a countable alpha-Toronto space for each alpha<=omega. We also construct consistent examples of countable alpha-Toronto spaces for each alpha<omega_1.
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Computability of Separation Axioms in Countable Second Countable Spaces
本文直接计算拓扑指标集复杂性，为各 Tychonoff 分离公理及 Polish 性、Cantor–Bendixson 秩等性质确定精确算术完备性层级，弥补逆数学精度不足。
Some results on the $π$-weight of countable Fréchet-Urysohn spaces
本文研究了可数弗雷歇-尤里绍恩空间的π-重量谱，在标准米勒模型和随机实数模型中确定了其结果。
Assuming the Continuum Hypothesis, there is a compact first countable connected space of weight aleph_1 with no totally disconnected perfect subsets. Each such space, however, may be destroyed by some proper forcing order which does not add reals.
We show that, for a coanalytic subspace $X$ of $2^ω$, the countable dense homogeneity of $X^ω$ is equivalent to $X$ being Polish. This strengthens a result of Hrušák and Zamora Avilés. Then, inspired by results of Hernández-Gutiérrez, Hrušák and van Mill, using a technique of Medvedev, we construct a non-Polish subspace $X$ of $2^ω$ such that $X^ω$ is countable dense homogeneous. This gives the first $\mathsf{ZFC}$ answer to a question of Hrušák and Zamora Avilés. Furthermore, since our example is consistently analytic, the equivalence result mentioned above is sharp. Our results also answer a…
Countable spaces, realcompactness, and the pseudointersection number
本文研究可数空间的实紧嵌入维数 Exp(X)，证明存在可数拥挤空间 X 使 Exp(X)=κ 当且仅当伪交集数 p≤κ≤连续统 c，并借助第一可数空间乘积的可数子集伪特征分析及散射空间紧化结论，给出任意可数空间 Exp(X) 的计算。
We answer several questions of V. Tkačuk from [Point-countable $π$-bases in first countable and similar spaces, Fund. Math. 186 (2005), pp. 55--69.] by showing that (1) there is a ZFC example of a first countable, 0-dimensional Hausdorff space with no point-countable $π$-base (in fact, the order of any $π$-base of the space is at least $\aleph_ω$); (2) if there is a $κ$-Suslin line then there is a first countable GO space of cardinality $κ^+$ in which the order of any $π$-base is at least $κ$; (3) it is consistent to have a first countable, hereditarily Lindel\" of regular space having uncount…
We show (in ZFC) that the cardinality of a compact homogeneous space of countable tightness is no more than the size of the continuum.
We investigate some versions of $d$-space, well-filtered space and Rudin space concerning various countability properties. The main results include: (i) if the sobrification of a $T_0$ space $X$ is first-countable, then $X$ is an $ω$-Rudin space; (ii) every $ω$-well-filtered space is sober if its sobrification is first-countable; (iii) if a $T_0$ space is second-countable or first-countable and with a countable underlying set, then it is a $ω$-Rudin space; (iv) every first-countable $T_0$ space is well-filtered determined; (v) every irreducible closed subset in a first-countable $ω$-well-filte…
Coincidence of the Dimensions of First Countable Spaces with a Countable Network
本文证明了第一可数仿紧σ-空间中大归纳维数Ind与覆盖维数dim相等，从而正面回答了Arkhangel'skii关于具有可数网络的第一可数空间三类维数是否一致的问题。
The main aim of the article is to show, in the absence of the Axiom of Choice, relationships between the following, independent of $\mathbf{ZF}$, statements: "Every countable product of compact metrizable spaces is separable (respectively, compact)" and "Every countable product of compact metrizable spaces is metrizable". Statements related to the above-mentioned ones are also studied. Permutation models (among them new ones) are shown in which a countable sum (also a countable product) of metrizable spaces need not be metrizable, countable unions of countable sets are countable and there is a…
Nagata conjectured that every $M$-space is homeomorphic to a closed subspace of the product of a countably compact space and a metric space. This conjecture was refuted by Burke and van Douwen, and A. Kato, independently. However, we can show that there is a c.c.c. poset $P$ of size $2^ω$ such that in $V^P$ Nagata's conjecture holds for each first countable regular space from the ground model (i.e. if a first countable regular space $X\in V$ is an $M$-space in $V^P$ then it is homeomorphic to a closed subspace of the product of a countably compact space and a metric space in $V^P$). In fact, w…
Let T be the family of open subsets of a topological space (not necessarily Hausdorff or even T_0). We prove that if T has a countable base and is not countable, then T has cardinality at least continuum.
Countable dense homogeneity in large products of Polish spaces
该研究统一处理波兰空间乘积的可数稠密齐性问题，重点关注不可数乘积，给出少于$\mathfrak{p}$个波兰空间乘积满足可数稠密齐性的条件，得到新应用并证明结果最优。
We prove that the free locally convex space $L(X)$ over a metrizable space $X$ has countable tightness if and only if $X$ is separable.
We show in detail that every compact countable subset of a metric space is homeomorphic to a countable ordinal number, which extends a result given by Mazurkiewicz and Sierpinski for finite-dimensional Euclidean spaces. In order to achieve this goal, we use Transfinite Induction to construct a specific homeomorphism. In addition, we prove that for all metric space $(E,d)$, the cardinality of the set of all the equivalence classes $\mathscr{K}_E$, up to homeomorphisms, of compact countable subsets of $E$ is less than or equal to $\aleph_1$, i.e. $|\mathscr{K}_E| \le \aleph_1$. We also show that…
A symmetrizability criterion of Arhangelskii implies that a second-countable Hausdorff space is symmetrizable if and only if it is perfect. We present an example of a non-symmetrizable second-countable submetrizable space of cardinality $\mathfrak q_0$ and study the smallest possible cardinality $\mathfrak q_i$ of a non-symmetrizable second-countable $T_i$-space for $i\in\{1,2\}$.
Infima and cardinal characteristics of critical ideals for countable compact spaces
本文研究了可数紧空间临界理想conv_α的结构，定义了具有特定性质的理想conv_{<α}，并探讨了这些理想在递减层次中的交织关系及相关基数不变量。
Michael asked whether every productively Lindelöf space is powerfully Lindelöf. Building of work of Alster and De la Vega, assuming the Continuum Hypothesis, we show that every productively Lindelöf space of countable tightness is powerfully Lindelöf. This strengthens a result of Tall and Tsaban. The same methods also yield new proofs of results of Arkhangel'skii and Buzyakova. Furthermore, assuming the Continuum Hypothesis, we show that a productively Lindelöf space $X$ is powerfully Lindelöf if every open cover of $X^ω$ admits a point-continuum refinement consisting of basic open sets. This …
We give a combinatorial characterization of countable submaximal subspaces of $2^κ$. Using a parametrized version of Mathias forcing, we prove that there exists a countable submaximal subspace of $2^{ω_1}$ whilst $\mathfrak{c}=ω_2$. Combining this with previous results, we construct a disjointly tight countable irresolvable space of weight $<\mathfrak{c}$, answering a question of Bella and Hrušák.
Let $\mathbb{A}$ denote the Alexandroff-Urysohn double arrow space. We prove the following results: (a) $\mathbb{A}\times{}^ω{2}$ is not countable dense homogeneous; (b) ${}^ω{\mathbb{A}}$ is not countable dense homogeneous; (c) $\mathbb{A}$ has exactly $\mathfrak{c}$ types of countable dense subsets. These results answer questions by Arhangel'ski\uı, Hrušák and van Mill.
Every $T_1$ connected first-countable space is a continuous open image of a connected metrizable space
本文回答了 Tkachuk 的问题，证明任意 $T_1$ 连通第一可数空间都可表示为某个连通可度量化空间在连续开映射下的像。
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.
We show that a Tychonoff discretely star-Lindelof space can have arbitrarily big extent and note that there are consistent examples of normal discretely star-Lindelof spaces with uncountable extent.
In this paper we consider the question of when the space $C_p(X)$ of continuous real-valued functions on $X$ with the pointwise convergence topology is countable dense homogeneous. In particular, we focus on the case when $X$ is countable with a unique non-isolated point $\infty$. In this case, $C_p(X)$ is countable dense homogeneous if and only if the filter of open neighborhoods of $\infty$ is a non-meager $P$-filter.
We define the direct sum of a countable family of pointed metric spaces in a way resembling the direct sum of groups. Then we prove that if a family consists of $0$-hyperbolic (in the sense of Gromov) and $D$-discrete spaces, then its direct sum has asymptotic property C. The main example is a countable direct sum of free groups of (possibly varying) finite rank. This is a generalization of T. Yamauchi's result concernig the countable direct sum of the integers.
This paper addresses several questions of Feng, Gruenhage, and Shen which arose from Michael's theory of continuous selections from countable spaces. We construct an example of a space which is $L$-selective but not $\mathbb{Q}$-selective from $\mathfrak{d}=ω_1$, and an $L$-selective space which is not selective for a $P$-point ultrafilter from the assumption of $\mathsf{CH}$. We also produce $\mathsf{ZFC}$ examples of Fréchet spaces where countable subsets are first countable which are not $L$-selective.
A cuf space (set, resp.) is a space (set, resp.) which is a countable union of finite subspaces (subsets, resp.). It is proved in $\mathbf{ZF}$ (with the absence of the axiom of choice) that all countable unions of cuf (denumerable, resp.) sets are cuf sets iff all countable products of cofinite cuf (denumerable, resp.) spaces are quasi-metrizable iff all countable products of one-point Hausdorff compactifications of infinite cuf (denumerable, resp.) spaces are quasi-metrizable. A countable product of one-point Hausdorff compactifications of denumerable discrete spaces is first-countable iff i…
The second author has recently shown ([20]) that any selectively (a) almost disjoint family must have cardinality strictly less than $2^{\alpeh_0}$, so under the Continuum Hypothesis such a family is necessarily countable. However, it is also shown in the same paper that $2^{\alpeh_0} < 2^{\alpeh_1}$ alone does not avoid the existence of uncountable selectively (a) almost disjoint families. We show in this paper that a certain effective parametrized weak diamond principle is enough to ensure countability of the almost disjoint family in this context. We also discuss the deductive strength of t…
A Hausdorff topological space $X$ is called $\textit{superconnected}$ (resp. $\textit{coregular}$) if for any nonempty open sets $U_1,\dots U_n\subseteq X$, the intersection of their closures $\bar U_1\cap\dots\cap\bar U_n$ is not empty (resp. the complement $X\setminus (\bar U_1\cap\dots\cap\bar U_n)$ is a regular topological space). A canonical example of a coregular superconnected space is the projective space $\mathbb Q\mathsf P^\infty$ of the topological vector space $\mathbb Q^{<ω}=\{(x_n)_{n\inω}\in \mathbb Q^ω:|\{n\inω:x_n\ne 0\}|<ω\}$ over the field of rationals $\mathbb Q$. The space…
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。
We study some variations of the product topology on families of clopen subsets of $2^{\mathbb{N}}\times\mathbb{N}$ in order to construct countable nodec regular spaces (i.e. in which every nowhere dense set is closed) with analytic topology which in addition are not selectively separable and do not satisfy the combinatorial principle $q^+$.
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.
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.
As it was introduced by Tkachuk and Wilson, a topological space $X$ is cellular-compact if given any cellular, i.e. disjoint, family $\mathcal U$ of non-empty open subsets of $X$ there is a compact subspace $K\subset X$ such that $K\cap U\ne \emptyset$ for each $U\in \mathcal U$. Answering several questions raised by Tkachuk and Wilson we show that (1) any first countable cellular-compact $T_2$ space is $T_3$, and so its cardinality is at most $\mathfrak{c} = 2^ω$; (2) $cov(\mathcal M)>ω_1$ implies that every first countable and separable cellular-compact $T_2$ space is compact; (3 if there is…
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…
We consider the compact spaces sigma_n(I) of subsets of an uncountable set I of cardinality at most n and their countable products. We give a complete classification of their Banach spaces of continuous functions and a partial topological classification.
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。
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, …
Given a dynamical system $(X,f)$, we let $E(X,f)$ denote its Ellis semigroup and $E(X,f)^* = E(X,f) \setminus \{f^n : n \in \mathbb{N}\}$. We analyze the Ellis semigroup of a dynamical system having a compact metric countable space as a phase space. We show that if $(X,f)$ is a dynamical system such that $X$ is a compact metric countable space and every accumulation point $X'$ is periodic, then either each function of $E(X,f)^*$ is continuous or each function of $E(X,f)^*$ is discontinuous. We describe an example of a dynamical system $(X,f)$ where $X$ is a compact metric countable space, the …
Closed discrete bases in countable-dimensional topological vector spaces
本文证明了任意可数拓扑向量空间（或等价地，任意可数素数指数阿贝尔拓扑群）都存在一个闭离散基。
Three small results on normal first countable linearly H-closed spaces
本研究借助PFA、MA$_{ω_1}$(S)[S]与PFA(S)[S]的已知拓扑推论，证明在对应模型下，带多种附加性质的正规第一可数线性H-闭空间均为紧空间，补充了相关拓扑空间的紧性结论。
It is shown that CH implies the existence of a compact Hausdorff space that is countable dense homogeneous, crowded and does not contain topological copies of the Cantor set. This contrasts with a previous result by the author which says that for any crowded Hausdorff space $X$ of countable $π$-weight, if ${}^ω{X}$ is countable dense homogeneous, then $X$ must contain a topological copy of the Cantor set.
A topological space $L$ is called a linear ordered topological space (LOTS) whenever there is a linear order $\leq$ on $L$ such that the topology on $L$ is generated by the open sets of the form $(a, b)$ with $a < b$ and $a, b \in L \cup \{ -\infty, +\infty \}$. A topological space $X$ is called a generalized ordered space (GO-space) whenever $X$ is topologically embeddable in a LOTS. Main Theorem: Let $X$ be a Hausdorff topological space. Assume that any continuous image of $X$ is a GO-space. Then $X$ is homeomorphic to a countable successor ordinal (with the order topology). The converse tri…
We study M-separability as well as some other combinatorial versions of separability. In particular, we show that the set-theoretic hypothesis b=d implies that the class of selectively separable spaces is not closed under finite products, even for the spaces of continuous functions with the topology of pointwise convergence. We also show that there exists no maximal M-separable countable space in the model of Frankiewicz, Shelah, and Zbierski in which all closed P-subspaces of w^* admit an uncountable family of nonempty open mutually disjoint subsets. This answers several questions of Bella, B…
The study of the sobriety of Scott spaces has got an relative long history in domain theory. Lawson and Hoffmann independently proved that the Scott space of every continuous directed complete poset (usually called domain) is sober. Johnstone constructed the first directed complete poset whose Scott space is non-sober. Not long after, Isbell gave a complete lattice with non-sober Scott space. Based on Isbell's example, Xu, Xi and Zhao showed that there is even a complete Heyting algebra whose Scott space is non-sober. Achim Jung then asked whether every countable complete lattice has a sober S…
Countable tightness is not discretely reflexive in $σ$-compact spaces
本文构造了多个$\sigma$-紧致空间（部分在ZFC中，部分在附加公理下），它们本身不满足可数紧致性，但所有离散子集的闭包却满足可数紧致性，甚至第一可数，从而否定了可数紧致性的离散自反性。
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 paratopological group $AP(X)$ are of countable type if and only if $X$ is discrete; (2) For a functionally Hausdorff space $X$, if the free abelian paratopological group $AP(X)$ is of subcountable type then $X$ has countable pseudocharacter. Moreover, …
Being motivated by the study of the space $C_c(X)$ of all continuous real-valued functions on a Tychonoff space $X$ with the compact-open topology, we introduced in [15] the concepts of a $cp$-network and a $cn$-network (at a point $x$) in $X$. In the present paper we describe the topology of $X$ admitting a countable $cp$- or $cn$-network at a point $x\in X$. This description applies to provide new results about the strong Pytkeev property, already well recognized and applicable concept originally introduced by Tsaban and Zdomskyy [43]. We show that a Baire topological group $G$ is metrizable…
We answer a question of Yasui. Morever, we show that if a Tychonoff space Y is countably 1-paracompact in every Tychonoff space X that contains Y as a closed subspace then Y is linearly Lindelof.
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…
In the absence of the axiom of choice, new results concerning sequential, Fréchet-Urysohn, $k$-spaces, very $k$-spaces, Loeb and Cantor completely metrizable spaces are shown. New choice principles are introduced. Among many other theorems, it is proved in $\mathbf{ZF}$ that every Loeb, $T_3$-space having a base expressible as a countable union of finite sets is a metrizable second-countable space whose every $F_σ$-subspace is separable; moreover, every $G_δ$-subspace of a second-countable, Cantor completely metrizable space is Cantor completely metrizable, Loeb and separable. It is also notic…
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.
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。
We consider 9 natural tightness conditions for topological spaces that are all variations on countable tightness and investigate the interrelationships between them. Several natural open problems are raised.
Finite approximations of countable metric and ultrametric compacta
通过改编同伦重构定理，证明可数度量或超度量紧空间可由有限 $T_0$ 空间序列的逆极限进行拓扑重构。
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…
A countable metric space whose Hoare power space is not co-sober
本文构造了一个可数零维度量空间，其Hoare幂空间不是co-sober，从而否定了Xu及He和Zhao的相关猜想，并建立了Hoare幂空间co-sober性与Scott空间sober性的等价条件。
We show in ZF that: (i) Every subcompact metrizable space is completely metrizable, and every completely metrizable space is countably subcompact. (ii) A metrizable space X=(X,T) is countably compact iff it is countably subcompact relative to T. (iii) For every metric space X=(X,d), X is compact iff it is subcompact relative to T. We also show: (iv) The negation of each of the statements, (a) every countably subcompact metrizable space is completely metrizable, (b) every countably subcompact metrizable space is subcompact, (c) every complete metrizable space is subcompact is relatively consist…
We study the class of first-countable Lindelöf scattered spaces, or "FLS" spaces. While every $T_3$ FLS space is homeomorphic to a scattered subspace of $\mathbb Q$, the class of $T_2$ FLS spaces turns out to be surprisingly rich. Our investigation of these spaces reveals close ties to $Q$-sets, Lusin sets, and their relatives, and to the cardinals $\mathfrak{b}$ and $\mathfrak{d}$. Many natural questions about FLS spaces turn out to be independent of $\mathsf{ZFC}$. We prove that there exist uncountable FLS spaces with scattered height $ω$. On the other hand, an uncountable FLS space with fin…
Notes on countable frames
本文研究了可数框架的连续性与谱的拟连续性关系，并探讨了 T₁ 空间和 Scott 空间对问题解答的影响。
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