Measurable cardinals and the cardinality of Lindelöf spaces

林德洛夫散射P-空间的基数序列

Cardinal Sequences of Lindelöf scattered P-spaces

该研究继续探究LLSP空间的基数序列，提出了从锥系统和特定偏序构造LLSP空间的方法，确定了其基数序列的限制，并给出了序列$\langle κ_α: α< ω_2 \rangle$作为LLSP空间基数序列的必要条件与充分条件。

A note on $δ$-strongly compact cardinals

In this paper we investigate more characterizations and applications of $δ$-strongly compact cardinals. We show that, for a cardinal $κ$ the following are equivalent: (1) $κ$ is $δ$-strongly compact, (2) For every regular $λ\ge κ$ there is a $δ$-complete uniform ultrafilter over $λ$, and (3) Every product space of $δ$-Lindelöf spaces is $κ$-Lindelöf. We also prove that in the Cohen forcing extension, the least $ω_1$-strongly compact cardinal is a precise upper bound on the tightness of the products of two countably tight spaces.

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…

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…

Cardinal Invariants for the $G_δ$ topology

We prove upper bounds for the spread, the Lindelöf number and the weak Lindelöf number of the $G_δ$-topology on a topological space and apply a few of our bounds to give a short proof to a recent result of Juhász and van Mill regarding the cardinality of a $σ$-countably tight homogeneous compactum.

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。

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…

论函子性 Lindelöf 化

On Functorial Lindelöfifiability

本文证明拓扑空间具有函子性 Lindelöf 化当且仅当其实紧化为 Lindelöf，并引入函子性 P-化的显式构造，进而研究离散空间与序数空间上的函子性 κ-Lindelöf 化可能性。

Generic left-separated spaces and calibers

We use a natural forcing to construct a left-separated topology on an arbitrary cardinal kappa. The resulting left-separated space X_kappa is also 0-dimensional T_2, hereditarily Lindelof, and countably tight. Moreover if kappa is regular then d(X_kappa)= kappa, hence kappa is not a caliber of X_kappa, while all other uncountable regular cardinals are. We also prove it consistent that for every countable set A of uncountable regular cardinals there is a hereditarily Lindelof T_3 space X such that rho=cf(rho)>omega is a caliber of X exactly if rho not in A.

Topological density of ccc Boolean algebras - every cardinality occurs.

For every uncountable cardinal mu there is a ccc Boolean algebra whose topological density is mu .

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。

On the cardinality and weight spectra of compact spaces, II

Let B(kappa, lambda) be the subalgebra of P(kappa) generated by [kappa]^{<= lambda}. It is shown that if B is any homomorphic image of B(kappa, lambda) then either |B|< 2^lambda or |B|=|B|^lambda, moreover if X is the Stone space of B then either |X| <= 2^{2^lambda} or |X|=|B|=|B|^lambda. This implies the existence of 0-dimensional compact T_2 spaces whose cardinality and weight spectra omit lots of singular cardinals of ``small'' cofinality.

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…

First-countable Lindelöf scattered spaces

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…

Antichains in products of linear orders

1. For many regular cardinals lambda (in particular, for all successors of singular strong limit cardinals, and for all successors of singular omega-limits), for all n in {2,3,4, ...} : There is a linear order L such that L^n has no (incomparability-)antichain of cardinality lambda, while L^{n+1} has an antichain of cardinality lambda . 2. For any nondecreasing sequence (lambda2,lambda3, ...) of infinite cardinals it is consistent that there is a linear order L such that L^n has an antichain of cardinality lambda_n, but not one of cardinality lambda_n^+ .

On removing one point from a compact space

If B is a compact space and B\{pt} is Lindelof then B^k\{pt} is star-Linedlof for every cardinality k. If B\{pt} is compact then B^k\{pt} is discretely star-Lindelof. In particular, this gives new examples of Tychonoff discretely star-Lindelof spaces with unlimited extent.

Extremally Disconnected Groups of Measurable Cardinality

Given an arbitrary measurable cardinal $κ$, a nondiscrete Hausdorff extremally disconnected topological group of cardinality $κ$ is constructed.

Resolvability and complete accumulation points

We prove that: I. For every regular Lindelöf space $X$ if $|X|=Δ(X)$ and $\mathrm{cf}|X|\neω$, then $X$ is maximally resolvable; II. For every regular countably compact space $X$ if $|X|=Δ(X)$ and $\mathrm{cf}|X|=ω$, then $X$ is maximally resolvable. Here $Δ(X)$, the dispersion character of $X$, is the minimum cardinality of a nonempty open subset of $X$. Statements I and II are corollaries of the main result: for every regular space $X$ if $|X|=Δ(X)$ and every set $A\subseteq X$ of cardinality $\mathrm{cf}|X|$ has a complete accumulation point, then $X$ is maximally resolvable. Moreover, regu…

Weakly linearly Lindelöf monotonically normal spaces are Lindelöf

We call a space $X$ {\it weakly linearly Lindelöf} if for any family $\mathcal{U}$ of non-empty open subsets of $X$ of regular uncountable cardinality $κ$, there exists a point $x\in X$ such that every neighborhood of $x$ meets $κ$-many elements of $\mathcal{U}$. We also introduce the concept of {\it almost discretely Lindelöf} spaces as the ones in which every discrete subspace can be covered by a Lindelöf subspace. We prove that, in addition to linearly Lindelöf spaces, both weakly Lindelöf spaces and almost discretely Lindelöf spaces are weakly linearly Lindelöf. The main result of the pape…

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

In this paper we continue to study of properties of $S(n)$-spaces. We establish bounded on the cardinality of $S(n)$-spaces.

On cardinality bounds for $θ^n$-Urysohn spaces

We introduce the class of $θ^{n}$-Urysohn spaces and the $n$-$θ$-closure operator. $θ^n$-Urysohn spaces generalize the notion of a Urysohn space. We estabilish bounds on the cardinality of these spaces and cardinality bounds if the space is additionally homogeneous.

小基数空间乘积的可分辨性

Resolvability in products of spaces of small cardinality

本文证明了两个小基数正则等同空间的乘积是ω-可分辨的，以及n+2个小基数Hausdorff等同空间的乘积也是ω-可分辨的。

On the cardinality of separable pseudoradial spaces

The aim of this paper is to consider questions concerning the possible maximum cardinality of various separable pseudoradial (in short: SP) spaces. The most intriguing question here is if there is, in ZFC, a regular (or just Hausdorff) SP of cardinality greater than $\mathfrak c$. While this question is left open, we establish a number of non-trivial results that we list: 1. It is consistent with Martin's Axiom and $\mathfrak c =\aleph_2$ that there is a countably tight and compact SP of cardinality $2^{\mathfrak c}$. 2. If $κ$ is a measurable cardinal then in the forcing extension obtained by…

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.

Cardinality Estimations of Sets with Interval Uncertainties in Finite Topological Spaces

In this paper, we have established boundaries of cardinal numbers of nonempty sets in finite non-$T_1$ topological spaces using interval analysis. For a finite set with known cardinality, we give interval estimations based on the closure and interior of the set. In this paper, we give new results for the cardinalities of non-empty semi-open sets in non-$T_1$ topological spaces as well as in extremely disconnected and hyperconnected topological spaces.

On the cardinality of $π(δ)$

We prove that the cardinality of transitive quasi-uniformities in a quasi-proximity class is at least $2^{2^{\aleph_0}}$ if there exist at least two transitive quasi-uniformities in the class. The transitive elements of $π(δ)$ are characterized if ${\cal V}_δ$ is transitive, and in this case we give a condition when there exists a unique transitive quasi-uniformity in $π(δ)$.

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}.

On the cardinality of Hausdorff spaces

A common generalization for two of the main streams of cardinality inequalities is developed; each stream derives from the famous inequality established by A.V. Arhangel'ski\uı in 1969 for Hausdorff spaces. At the end of one stream is the recent inequality by Bella and at the end of the second stream is the 1988 inequality by Bella and Cammaroto. This generalization is extended and used to analyze a result containing an increasing chain of spaces that satisfies the same cardinality inequality. The paper is concluded with some open problems.

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…

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。