Number of open sets for a topology with a countable basis
Saharon Shelah
设T是拓扑空间的开子集族（不一定是Hausdorff或T_0）。证明了如果T有可数基且不可数，则T至少有连续势。
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
Every transformation monoid comes equipped with a canonical topology-the topology of pointwise convergence. For some structures, the topology of the endomorphism monoid can be reconstructed from its underlying abstract monoid. This phenomenon is called automatic homeomorphicity. In this paper we show that whenever the automorphism group of a countable saturated structure has automatic homeomorphicity and a trivial center, then the monoid of elementary self-embeddings has automatic homeomorphicity, too. As a second result we strengthen a result by Lascar by showing that whenever $\mathbf{A}$ is…
Using the property of being completely Baire, countable dense homogeneity and the perfect set property we will be able, under Martin's Axiom for countable posets, to distinguish non-principal ultrafilters on $ω$ up to homeomorphism. Here, we identify ultrafilters with subpaces of $2^ω$ in the obvious way. Using the same methods, still under Martin's Axiom for countable posets, we will construct a non-principal ultrafilter $\UU\subseteq 2^ω$ such that $\UU^ω$ is countable dense homogeneous. This consistently answers a question of Hrušák and Zamora Avilés. Finally, we will give some partial resu…
Countable Choice and Compactness
我们在集合论中没有选择ZF。用AC（N）表示可数选择选择公理，证明了在ZF+AC（N）中一致凸巴拿赫空间的闭单位球在凸拓扑（ZF中的弱拓扑的一种替代）中是紧的.我们证明了这个球在凸拓扑中是（紧）凸紧的。给定一个集合I，一个大于或等于1的实数p（分别为. p = 0）和[0，1]^I的某个闭子集F（它是l^p（I）的有界子集），我们证明了AC（N）（resp. DC，依赖选择公理）蕴涵了F的紧性。
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…
We introduce a topology on the space of all isomorphism types represented in a given class of countable models, and use this topology as an aid in classifying the isomorphism types. This mixes ideas from effective descriptive set theory and computable structure theory, extending concepts from the latter beyond computable structures to examine the isomorphism problem on arbitrary countable structures. We give examples using specific classes of fields and of trees, illustrating how the new concepts can yield classifications that reveal differences between seemingly similar classes. Finally, we u…
Topology, forcing, and graph colourings
本文提出一种forcing方法，用于证明某些图无法具有可数着色的Borel类alpha。构造了对这类着色弱最小的图。
No Countable Basis for Borel Directed Graphs of Dichromatic Number at Least Three
本文证明了Borel有向图中顶点可被分成两个Borel无环集的图类为Σ¹₂完全集，表明判断Borel有向图的二色数至少为三是一个Π¹₂完全问题。
$σ$-locales in Formal Topology
本文证明$\sigma$-框架（即$\sigma$-locale）可纳入形式拓扑框架，并构造性地刻画最小强稠密$\sigma$-子locale，给出布尔locale的$\sigma$-版本，依赖可数选择公理。
Scott products and sobriety of countable meet-continuous dcpos
本文研究可数交连续 dcpo 的 Scott 积与 sobriety，构造反例否定两个问题，并证明 L-dcpo 在 Scott 平方为普通乘积拓扑时是 Scott sober，同时刻画 well-filteredness。
We say that an ideal I is homogeneous, if its restriction to any I-positive subset is isomorphic to I. The paper investigates basic properties of this notion -- we give examples of homogeneous ideals and present some applications to topology and ideal convergence. Moreover, we answer questions related to our research.
An ideal on a set $X$ is a collection of subsets of $X$ closed under the operations of taking finite unions and subsets of its elements. Ideals are a very useful notion in topology and set theory and have been studied for a long time. We present a survey of results about ideals on countable sets and include many open questions.
We study the class of analytic binary relations on Polish spaces, compared with the notions of continuous reducibility or injective continuous reducibility. In particular, we characterize when a locally countable Borel relation is $Σ$ 0 $ξ$ (or $Π$ 0 $ξ$), when $ξ$ $\ge$ 3, by providing a concrete finite antichain basis. We give a similar characterization for arbitrary relations when $ξ$ = 1. When $ξ$ = 2, we provide a concrete antichain of size continuum made of locally countable Borel relations minimal among non-$Σ$ 0 2 (or non-$Π$ 0 2) relations. The proof of this last result allows us to s…
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。
In this paper, the class of all linearly ordered topological spaces (LOTS) quasi-ordered by the embeddability relation is investigated. In ZFC it is proved that for countable LOTS this quasi-order has both a maximal (universal) element and a finite basis. For the class of uncountable LOTS of cardinality $κ$ it is proved that this quasi-order has no maximal element for $κ$ at least the size of the continuum and that in fact the dominating number for such quasi-orders is maximal, i.e. $2^κ$. Certain subclasses of LOTS, such as the separable LOTS, are studied with respect to the top and internal …
Scott topologies on products of countable complete Heyting algebras
本文证明斯科特拓扑与可数表示框架的任意乘积相容且乘积空间是素谱的，并指出可数生成不足以保证该性质，同时给出阿廷空间与代数 dcpo 的关联及二分法定理。
We show that every basis for the countable Borel equivalence relations strictly above $\mathbb{E}_0$ under measure reducibility is uncountable, thereby ruling out natural generalizations of the Glimm-Effros dichotomy. We also push many known results concerning the abstract structure of the measure reducibility hierarchy to its base, using arguments substantially simpler than those previously employed.
Reverse mathematics of regular countable second countable spaces
研究正则可数第二可数空间特征定理的逆数学，证明算术理解与$T_3$ $CSCS$可度量化等等价，刻画相关空间，表明一些拓扑学结论与算术超限递归或$\Pi^1_1$理解等价。
We prove that the Fraïssé limit of a Fraïssé class $\mathcal C$ is the (unique) countable structure whose isomorphism type is comeager (with respect to a certain logic topology) in the Baire space of all structures whose age is contained in $\mathcal C$ and which are defined on a fixed countable universe. In particular, the set of groups isomorphic to Hall's universal group is comeager in the space of all countable locally finite groups and the set of fields isomorphic to the algebraic closure of $\mathbb F_p$ is comeager in the space of countable fields of characteristic $p$.
Every countable meet-continuous lattice is Scott sober
本文证明每个可数交连续格是Scott素朴的，并肯定回答Xu的问题7.1和7.2，通过对角线论证和并平移构造Scott开矩形，还刻画了非素朴格的基数谱。
We use tools of mathematical logic to analyse the notion of a path on an complex algebraic variety, and are led to formulate a "rigidity" property of fundamental groups specific to algebraic varieties, as well as to define a bona fide topology closely related to etale topology. These appear as criteria for uncountable categoricity, or rather stability and homogeneity, of the formal countable language we propose to describe homotopy classes of paths on a variety, or equivalently, its universal covering space. We also show that, with this topology, the universal covering space of the variety is …
Computable Bases
本文研究可计算基底和预基底，推广Kreitz-Weihrauch定理，探讨拓扑空间的可计算描述方法，并建立预基底与T_0空间表示之间的伽罗瓦连接。
Continuous colorings on compact spaces
本文研究零维可度量化紧空间上无连续着色的图类，证明其存在反链基并确定其大小，同时给出可数情形下的具体反链基，并讨论相关拟序与等价关系的解析完全性。
A compact topological space X is spectral if it is sober (i.e., every irreducible closed set is the closure of a unique singleton) and the compact open subsets of X form a basis of the topology of X, closed under finite intersections. Theorem. A topological space X is homeomorphic to the spectrum of some countable Abelian {\ell}-group with unit (resp., MV-algebra) iff X is spectral, has a countable basis of open sets, and for any points x and y in the closure of a singleton {z}, either x is in the closure of {y} or y is in the closure of {x}. We establish this result by proving that a countabl…
Hurewicz-like tests for Borel subsets of the plane
设xi是一个非空可数序数。我们研究的Borel子集的平面，可以使$\bormxi$通过细化波兰拓扑的实数线。这些集合称为潜在$\bormxi$。我们给出了一个类似Hurewicz的测试来识别潜在的$\bormxi$集。
In his PhD Thesis Konstantinos Beros proved a number of results about compactly generated subgroups of Polish groups. Such a group is K-sigma - the countable union of compact sets. He notes that the group of rationals under addition with the discrete topology is an example of a Polish group which is K-sigma (since it is countable) but not compactly generated. Beros showed that for any Polish group G, every K-sigma subgroup of G is compactly generated iff every countable subgroup of G is compactly generated. Beros showed that any K-sigma subgroup of Z^omega (infinite product of the integers) is…
Forcing tightness in products of fans
theta扇F_theta是通过将乘积theta \times（omega+1）的非孤立点与单个点\infty标识而获得的商空间。(Here，θ具有离散拓扑，而Ω +1具有有序拓扑。我们定义了几个强迫的概念，使我们能够操纵产品的紧密性。一些后果包括：不（F_theta \times F_omega）= theta并不意味着存在（θ，ω）-间隙，第一可数<θ-集合Hausdorff（cwH）空间的新例子，对于奇异基数θ，不是\leq θ-cwH，对于基数λ \leq θ，具有cf（θ）\geq ω_1和λ正则，第一可数<θ-cwH不是\leq θ-cwH空间，其可通过移除基数λ的闭合离散集而成为cwH。我们还证明了两个定理，刻画产品的整值函数族的球迷的紧密性。
We give a theoretical and applicable framework for dealing with real-world phenomena. Joining pointwise and pointfree notions in BISH, natural topology gives a faithful idea of important concepts and results in intuitionism. Natural topology is well-suited for practical and computational purposes. We give several examples relevant for applied mathematics, such as the decision-support system Hawk-Eye (used in professional tennis), and various real-number representations. We compare CLASS, INT, RUSS, BISH and formal topology. There are links with physics, regarding the topological character of o…
Let $ξ\geq 1$ be a countable ordinal. We study the Borel subsets of the plane that can be made ${\bfΠ}^0_ξ$ by refining the Polish topology on the real line. These sets are called potentially ${\bfΠ}^0_ξ$. We give a Hurewicz-like test to recognize potentially ${\bfΠ}^0_ξ$ sets.
We extend a dichotomy between 1-basedness and supersimplicity proved in a previous paper. The generalization we get is to arbitrary language, with no restrictions on the topology (we do not demand type-definabilty of the open set in the definition of essential 1-basedness). We conclude that every (possibly uncountable) hypersimple unidimensional theory that is not s-essentially 1-based by means of the forking topology is supersimple. We also obtain a strong version of the above dichotomy in the case where the language is countable.
Let $G$ be a topological group and let $μ$ be the Lebesgue measure on the interval $[0,1]$. We let $L_0(G)$ to be the topological group of all $μ$-equivalence classes of $μ$-measurable functions defined on [0,1] with values in $G$, taken with the pointwise multiplication and the topology of convergence in measure. We show that for a Polish group $G$, if $L_0(G)$ has ample generics, then $G$ has ample generics, thus the converse to a result of Kaïchouh and Le Maître. We further study topological similarity classes and conjugacy classes for many groups ${\rm{Aut}}(M)$ and $L_0({\rm{Aut}}(M))$, w…
Let $\mathrm{G}$ be a subgroup of the symmetric group $\mathfrak S(U)$ of all permutations of a countable set $U$. Let $\overline{\mathrm{G}}$ be the topological closure of $\mathrm{G}$ in the function topology on $U^U$. We initiate the study of the poset $\overline{\mathrm{G}}[U]:=\{f[U]\mid f\in \overline{\mathrm{G}}\}$ of images of the functions in $\overline{\mathrm{G}}$, being ordered under inclusion. This set $\overline{\mathrm{G}}[U]$ of subsets of the set $U$ will be called the \emph{poset of copies for} the group $\mathrm{G}$. A denomination being justified by the fact that for every …
A logic of co-valuations
本文提出一种基于共估值的逻辑，用于研究紧拓扑对象，展示了模型论工具和拓扑概念在该框架内的应用
Models of an Abstract Elementary Class as a Generalized Polish Space
本文利用 Baldwin-Boney 关系表示定理，将一阶逻辑中模型空间为波兰空间的结果推广到抽象初等类，证明其模型在特定条件下构成广义波兰空间。
Generic length functions on countable groups
该研究针对不满足非平凡混合恒等式的可数群，证明了Baire范畴意义下的generic长度函数是字长度函数且对应Cayley图同构于通用图U，同时得到剩余子集含2^ℵ₀个渐近不可比长度函数等结论，揭示了generic长度函数的模型论不可区分性。
Largeness and generalized t-henselianity
本文证明可数域在Pop意义下为大域当且仅当其容许广义t-亨塞尔域拓扑，并用所有gt-亨塞尔拓扑刻画了étale开拓扑。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。