On the cardinality of $S(n)$-spaces

Lindelof indestructibility, topological games and selection principles

Arhangel'skii proved that if a first countable Hausdorff space is Lindelöf, then its cardinality is at most $2^{\aleph_0}$. Such a clean upper bound for Lindelöf spaces in the larger class of spaces whose points are ${\sf G}_δ$ has been more elusive. In this paper we continue the agenda started in F.D. Tall, On the cardinality of Lindelöf spaces with points $G_δ$, Topology and its Applications 63 (1995), 21 - 38, of considering the cardinality problem for spaces satisfying stronger versions of the Lindelöf property. Infinite games and selection principles, especially the Rothberger property, a…

n-Hausdorff 空间与 n-Urysohn 空间基数的新界

New bounds on the cardinality of n-Hausdorff and n-Urysohn spaces

研究通过引入分别推广伪特征与闭伪特征的两个新基数函数，给出 n-Urysohn 空间基数的新界，并证明 n-Urysohn n-H-闭空间的相关性质。

On the cardinality of the $θ$-closed hull of sets

The θ-closed hull of a set A in a topological space is the smallest set C containing A such that, whenever all $closed$ neighborhoods of a point intersect C, this point is in C. We define a new topological cardinal invariant function, the $θ-bitighness small number$ of a space X, bts_theta(X), and prove that in every topological space X, the cardinality of the theta-closed hull of each set A is at most |A|^{bts_theta(X)}. Using this result, we synthesize all earlier results on bounds on the cardinality of theta-closed hulls. We provide applications to P-spaces and to the almost-Lindelof number…

On the cardinality of the $θ$-closed hull of sets II

The research in this paper is a continuation of the investigation of the cardinality of the $θ$-closed hull of subsets of spaces. This research obtains new upper bounds of the cardinality of the $θ$-closed hull of subsets using cardinal functions of $θ$-bitightness (\cite{Ca-Ko}), finite $θ$-bitightness (\cite{Ca-Ca-Pa-Ts}) and $θ$-bitightness small number (\cite{Ca-Ca-Pa-Ts}) of spaces. In the final section, examples of spaces are presented including one that answers a question posed in \cite{Bo-Ca-Ma} and \cite{Bo-Pa}.

Measurable cardinals and the cardinality of Lindelöf spaces

If it is consistent that there is a measurable cardinal, then it is consistent that all points g-delta Rothberger spaces have "small" cardinality.

Compact condensations of Hausdorff spaces

In this paper, we continue to study one of the classic problems in general topology raised by P.S. Alexandrov: when a Hausdorff space $X$ has a continuous bijection (a condensation) onto a compactum? We concentrate on the situation when not only $X$ but also $X\setminus Y$ can be condensed onto a compactum whenever the cardinality of $Y$ does not exceed certain $τ$.

On a game theoretic cardinality bound

The main purpose of the paper is the proof of a cardinal inequality for a space with points $G_δ$, obtained with the help of a long version of the Menger game. This result improves a similar one of Scheepers and Tall.

New bounds on the cardinality of Hausdorff spaces and regular spaces

Using weaker versions of the cardinal function $ψ_c(X)$, we derive a series of new bounds for the cardinality of Hausdorff spaces and regular spaces that do not involve $ψ_c(X)$ nor its variants at all. For example, we show if $X$ is regular then $|X|\leq 2^{c(X)^{πχ(X)}}$ and $|X|\leq 2^{c(X)πχ(X)^{ot(X)}}$, where the cardinal function $ot(X)$, introduced by Tkachenko, has the property $ot(X)\leq\min\{t(X),c(X)\}$. It follows from the latter that a regular space with cellularity at most $\mathfrak{c}$ and countable $π$-character has cardinality at most $2^\mathfrak{c}$. For a Hausdorff space …

Cardinality bounds involving the skew-$λ$ Lindelöf degree and its variants

We introduce a modified closing-off argument that results in several improved bounds for the cardinalities of Hausdorff and Urysohn spaces. These bounds involve the cardinal invariant $skL(X,λ)$, the skew-$λ$ Lindelöf degree of a space $X$, where $λ$ is a cardinal. $skL(X,λ)$ is a weakening of the Lindelöf degree and is defined as the least cardinal $κ$ such that if $\mathcal{U}$ is an open cover of $X$ then there exists $\mathcal{V}\in [\mathcal{U}]^{\leqκ}$ such that $|X\backslash\cup\mathcal{V}|<λ$. We show that if $X$ is Hausdorff then $|X|\leq 2^{skL(X,λ)t(X)ψ(X)}$, where $λ= 2^{t(X)ψ(X)}…

A survey of cardinality bounds on homogeneous topological spaces

In this survey we catalogue the many results of the past several decades concerning bounds on the cardinality of a topological space with homogeneous or homogeneous-like properties. These results include van Douwen's Theorem, which states $|X|\leq 2^{πw(X)}$ if $X$ is a power homogeneous Hausdorff space, and its improvements $|X|\leq d(X)^{πχ(X)}$ and $|X|\leq 2^{c(X)πχ(X)}$ for spaces $X$ with the same properties. We also discuss de la Vega's Theorem, which states that $|X|\leq 2^{t(X)}$ if $X$ is a homogeneous compactum, as well as its recent improvements and generalizations to other setting…

Cardinal invariants of cellular-Lindelof spaces

A space $X$ is said to be "cellular-Lindelöf" if for every cellular family $\mathcal{U}$ there is a Lindelöf subspace $L$ of $X$ which meets every element of $\mathcal{U}$. Cellular-Lindelöf spaces generalize both Lindelöf spaces and spaces with the countable chain condition. Solving questions of Xuan and Song, we prove that every cellular-Lindelöf monotonically normal space is Lindelöf and that every cellular-Lindelöf space with a regular $G_δ$-diagonal has cardinality at most $2^\mathfrak{c}$. We also prove that every normal cellular-Lindelöf first-countable space has cardinality at most con…

Regular $G_δ$-diagonals and some upper bounds for cardinality of topological spaces

We prove that, under CH, any space with a regular $G_δ$-diagonal and caliber $ω_1$ is separable; a corollary of this result answers, under CH, a question of Buzyakova. For any Urysohn space $X$, we establish the inequality $|X|\le wL(X)^{sΔ_2(X)\cdot{dot(X)}}$ which represents a generalization of a theorem of Basile, Bella, and Ridderbos. We also show that if $X$ is a Hausdorff space, then $|X|\le(πχ(X)\cdot d(X))^{ot(X)\cdotψ_c(X)}$; this result implies Šapirovski{\uı}'s inequality $|X|\leπχ(X)^{c(X)\cdotψ(X)}$ which only holds for regular spaces. It is also proved that $|X|\le πχ(X)^{ot(X)\c…

关于紧空间的势谱和权谱，II

On the cardinality and weight spectra of compact spaces, II

设B（kappa，lambda）是P（kappa）由[kappa]^{<= lambda}生成的子代数。证明了：若B是B（kappa，lambda）的任意同态像，则|B| <2 ^lambda或|B| = |B| ^lambda，且若X是B的Stone空间，则|X| <= 2^{2 ^lambda}或|X| = |B| = |B| ^lambda.这意味着存在0维紧T_2空间，其基数和权谱省略了大量“小”共尾性的奇异基数。

Cardinality of the Ellis semigroup on compact metric countable spaces

Let $E(X,f)$ be the Ellis semigroup of a dynamical system $(X,f)$ where $X$ is a compact metric space. We analyze the cardinality of $E(X,f)$ for a compact countable metric space $X$. A characterization when $E(X,f)$ and $E(X,f)^* = E(X,f) \setminus \{ f^n : n \in \mathbb{N}\}$ are both finite is given. We show that if the collection of all periods of the periodic points of $(X,f)$ is infinite, then $E(X,f)$ has size $2^{\aleph_0}$. It is also proved that if $(X,f)$ has a point with a dense orbit and all elements of $E(X,f)$ are continuous, then $|E(X,f)| \leq |X|$. For dynamical systems of th…

On Cardinality Of Non Isomorphic Intermediate Rings Of C(X)

Let $\sum (X)$ be the collection of subalgebras of $C(X)$ containing $C^{*}(X)$, where $X$ is a Tychonoff space. For any $A(X)\in \sum(X)$ there is associated a subset $\upsilon_{A}(X)$ of $βX$ which is an $A$-analogue of the Hewitt real compactification $\upsilon X$ of $X$. For any $A(X)\in \sum(X)$, let $[A(X)]$ be the class of all $B(X)\in \sum(X)$ such that $\upsilon_{A}(X)=\upsilon_{B}(X)$. We have shown that for first countable non compact real compact space $X$, $[A(X)]$ contains at least $2^{c}$ many different subalgebras no two of which are isomorphic.

On symmetrizability and perfectness of second-countable spaces

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 $i\in\{1,2\}$.

More on Cardinality Bounds Involving the Weak Lindelöf degree

We give several new bounds for the cardinality of a Hausdorff topological space $X$ involving the weak Lindelöf degree $wL(X)$. In particular, we show that if $X$ is extremally disconnected, then $|X|\leq 2^{wL(X)πχ(X)ψ(X)}$, and if $X$ is additionally power homogeneous, then $|X|\leq 2^{wL(X)πχ(X)}$. We also prove that if $X$ is an almost Lindelöf space with a strong $G_δ$-diagonal of rank 2, then $|X|\leq 2^{\aleph_0}$; that if $X$ is a star-cdc space with a $G_δ$-diagonal of rank 3, then $|X| \le 2^{\aleph_0}$; and if $X$ is any normal star-cdc space $X$ with a $G_δ$-diagonal of rank 2, the…

First countable and almost discretely Lindelöf $T_3$ spaces have cardinality at most continuum

A topological space $X$ is called almost discretely Lindelöf if every discrete set $D \subset X$ is included in a Lindelöf subspace of $X$. We say that the space $X$ is {\em $μ$-sequential} if for every non-closed set $A \subset X$ there is a sequence of length $\le μ$ in $A$ that converges to a point which is not in $A$. With the help of a technical theorem that involves elementary submodels, we establish the following two results concerning such spaces. (1) For every almost discretely Lindelöf $T_3$ space $X$ we have $|X| \le 2^{χ(X)}$. (2) If $X$ is a $μ$-sequential $T_2$ space of pseudocha…

On cardinality bounds involving the weak Lindelöf degree

We give a general closing-off argument in Theorem 2.1 from which several corollaries follow, including (1) if $X$ is a locally compact Hausdorff space then $|X|\leq 2^{wL(X)ψ(X)}$, and (2) if $X$ is a locally compact power homogeneous Hausdorff space then $|X|\leq 2^{wL(X)t(X)}$. The first extends the well-known cardinality bound $2^{ψ(X)}$ for a compactum $X$ in a new direction. As $|X|\leq 2^{wL(X)χ(X)}$ for a normal space $X$ [3], this enlarges the class of known Tychonoff spaces for which this bound holds. In 2.10 we give a short, direct proof of (1) that does not use 2.1. Yet 2.1 is broad…

On $σ$-countably tight spaces

Extending a result of R. de la Vega, we prove that an infinite homogeneous compactum has cardinality $\mathfrak{c}$ if either it is the union of countably many dense or finitely many arbitrary countably tight subspaces. The question if every infinite homogeneous and $σ$-countably tight compactum has cardinality $\mathfrak{c}$ remains open. We also show that if an arbitrary product is $σ$-countably tight then all but finitely many of its factors must be countably tight.

Infinite games and cardinal properties of topological spaces

Inspired by work of Scheepers and Tall, we use properties defined by topological games to provide bounds for the cardinality of topological spaces. We obtain a partial answer to an old question of Bell, Ginsburg and Woods regarding the cardinality of weakly Lindelöf first-countable regular spaces and answer a question recently asked by Babinkostova, Pansera and Scheepers. In the second part of the paper we study a game-theoretic version of cellularity, a special case of which has been introduced by Aurichi. We obtain a game-theoretic proof of Shapirovskii's bound for the number of regular open…

On the cardinality of Hausdorff spaces and H-closed spaces

We introduce the cardinal invariant $aL^\prime(X)$ and show that $|X|\leq 2^{aL^\prime(X)χ(X)}$ for any Hausdorff space $X$ (a corollary of Theorem 4.4. This invariant has the properties a) $aL^\prime(X)=\aleph_0$ if $X$ is H-closed, and b) $aL(X)\leq aL^\prime(X)\leq aL_c(X)$. Theorem 4.4 then gives a new improvement of the well-known Hausdorff bound $2^{L(X)χ(X)}$ from which it follows that $|X|\leq 2^{ψ_c(X)}$ if $X$ is H-closed (Dow/Porter [5]). The invariant $aL^\prime(X)$ is constructed using convergent open ultrafilters and an operator $c:\scr{P}(X)\to\scr{P}(X)$ with the property $clA\…

Hausdorff SDL空间的基数界

Cardinality Bounds for Hausdorff SDL Spaces

本文针对Hausdorff SDL空间建立基数界不等式，推导得第一可数该类空间基数不超过连续统，解答了Bella与Spadaro的两个问题，还给出SDL空间的一致有界分解性质相关中间结果。

有限超度量空间与完美二叉树的距离中心极大化

Maximal Center of Distances of Finite Ultrametric Spaces and Perfect Binary Trees

本文研究有限超度量空间中距离中心基数与集合基数的最小化关系，证明等距性等价于距离中心相同，并刻画了表示树的同构类型。

Large strongly anti-Urysohn spaces exist

As defined in [1], a Hausdorff space is strongly anti-Urysohn (in short: SAU) if it has at least two non-isolated points and any two infinite} closed subsets of it intersect. Our main result answers the two main questions of [1] by providing a ZFC construction of a locally countable SAU space of cardinality $2^{\mathfrak{c}}$. The construction hinges on the existence of $2^{\mathfrak{c}}$ weak P-points in $ω^*$, a very deep result of Ken Kunen. It remains open if SAU spaces of cardinality $> 2^{\mathfrak{c}}$ could exist, while it was shown in [1] that $2^{2^{\mathfrak{c}}}$ is an upper bound.…

任意 Hausdorff 空间密度的一个界

A bound for the density of any Hausdorff space

本文引入非准正则度 nq(X)，把 Sapirovskii 的密度界推广到任意 Hausdorff 空间，并统一推出密度与基数界及改进的基数上界。

关于T_3空间中基数的省略

On omitting cardinals in T_3 spaces

本文证明了对每个（无限基数）λ，存在一个具有闭基的T_3-空间X，其中2^mu点，其中mu = 2^lambda，使得基数< |X|的每个闭子空间的基数<λ.

Large widely-connected spaces

We show that there are widely-connected spaces of arbitrarily large cardinality, answering a question by David Bellamy.

Variations on known and recent cardinality bounds

Sapirovskii [18] proved that $|X|\leqπχ(X)^{c(X)ψ(X)}$, for a regular space $X$. We introduce the $θ$-pseudocharacter of a Urysohn space $X$, denoted by $ψ_θ(X)$, and prove that the previous inequality holds for Urysohn spaces replacing the bounds on celluarity $c(X)\leqκ$ and on pseudocharacter $ψ(X)\leqκ$ with a bound on Urysohn cellularity $Uc(X)\leqκ$ (which is a weaker conditon because $Uc(X)\leq c(X)$) and on $θ$-pseudocharacter $ψ_θ(X)\leqκ$ respectivly (note that in general $ψ(\cdot)\leqψ_θ(\cdot)$ and in the class of regular spaces $ψ(\cdot)=ψ_θ(\cdot)$). Further, in [6] the authors g…

Cardinalities of weakly Lindelöf spaces with regular $G_κ$-diagonals

For a Urysohn space $X$ we define the regular diagonal degree $\overlineΔ(X)$ of $X$ to be the minimal infinite cardinal $κ$ such that $X$ has a regular $G_κ$-diagonal i.e. there is a family $(U_η:η<κ)$ of open neighborhoods of $Δ_X=\{(x,x)\in X^2:x\in X\}$ in $X^2$ such that $Δ_X = \bigcap_{η<κ} \overline{U}_η$. In this paper we show that if $X$ is a Urysohn space then: (1) $|X|\leq 2^{c(X)\cdot\overlineΔ(X)}$; (2) $|X|\leq 2^{\overlineΔ(X)\cdot 2^{wL(X)}}$; (3) $|X|\le wL(X)^{\overlineΔ(X)\cdotχ(X)}$; and (4) $|X|\le aL(X)^{\overlineΔ(X)}$; where $χ(X)$, $c(X)$, $wL(X)$ and $aL(X)$ are respe…

$Q$-spaces, perfect spaces and related cardinal characteristics of the continuum

A topological space $X$ is called a $Q$-space if every subset of $X$ is of type $F_σ$ in $X$. For $i\in\{1,2,3\}$ let $\mathfrak q_i$ be the smallest cardinality of a second-countable $T_i$-space which is not a $Q$-space. It is clear that $\mathfrak q_1\le\mathfrak q_2\le\mathfrak q_3$. For $i\in\{1,2\}$ we prove that $\mathfrak q_i$ is equal to the smallest cardinality of a second-countable $T_i$-space which is not perfect. Also we prove that $\mathfrak q_3$ is equal to the smallest cardinality of a submetrizable space, which is not a $Q$-space. Martin's Axiom implies that $\mathfrak q_i=\mat…

关于图要么连通要么局部连通的实函数

On real functions with graphs either connected or locally connected

该研究分类了所有实数线的细化拓扑，证明了某些子空间的可嵌入性和可度量性，并揭示了局部连通空间的结构特性。

The minimal cardinality where the Reznichenko property fails

A topological space X$ has the Frechet-Urysohn property if for each subset A of X and each element x in the closure of A, there exists a countable sequence of elements of A which converges to x. Reznichenko introduced a natural generalization of this property, where the converging sequence of elements is replaced by a sequence of disjoint finite sets which eventually intersect all neighborhoods of x. In their paper, Kocinac and Scheepers conjecture that the minimal cardinality of a set X of real numbers such that C_p(X) does not have the weak Frechet-Urysohn property is equal to b. (b is the m…

The Stone-Cech compactifications of $ω^*\setminus \{x\}$ and $S_κ\setminus\{x\}$

The space $S_κ$ is the Stone space of the $κ$-saturated Boolean algebra of cardinality $κ$. It exists provided that $κ= κ^{<κ}$, and is characterised topologically as the unique $κ$-Parovichenko space of weight $κ$. Under the Continuum Hypothesis, $S_{ω_1}$ coincides with $ω^*$. This paper investigates questions related to the Stone-Cech compactification of spaces $S_κ\setminus \{x\}$, extending corresponding results obtained by Fine & Gillman and Comfort & Negrepontis for the space $ω^*$. We show that for every point $x$ of $S_κ$, the Stone-Cech remainder of $S_κ\setminus \{x\}$ is a $κ^+$-Pa…

Topology, Metric Spaces and the Generalized Continuum Hypothesis

This is a paper that aims to interpret the cardinality of a set in terms of Baire Category, i.e. how many closed nowhere dense sets can be deleted from a set before the set itself becomes negligible. . To do this natural tree-theoretic structures such as the Baire topology are introduced, and the Baire Category Theorem is extended to a statement that a $\aleph$-sequentially complete binary tree representation of a Hausdorff topological space that has a clopen base of cardinality $\aleph$ and no isolated or discrete points is not the union of $<\aleph+1$-many nowhere dense subsets for cardinal …

具有 Lindelöf 闭包的强离散子集

Strongly discrete subsets with Lindelöf closures

本文引入 SDL 空间概念，研究其基数不变量，证明 SDL Urysohn 空间与 SDL P-空间的基数上界，并部分回答了 cellular-Lindelöf 空间的基数问题。

Submaximal properties in (strongly) topological gyrogroups

A space $X$ is submaximal if any dense subset of $X$ is open. In this paper, we prove that every submaximal topological gyrogroup of non-measurable cardinality is strongly $σ$-discrete. Moreover, we prove that every submaximal strongly topological gyrogroup of non-measurable cardinality is hereditarily paracompact.

The magnitude of metric spaces

Magnitude is a real-valued invariant of metric spaces, analogous to the Euler characteristic of topological spaces and the cardinality of sets. The definition of magnitude is a special case of a general categorical definition that clarifies the analogies between various cardinality-like invariants in mathematics. Although this motivation is a world away from geometric measure, magnitude, when applied to subsets of R^n, turns out to be intimately related to invariants such as volume, surface area, perimeter and dimension. We describe several aspects of this relationship, providing evidence for …