Some problems and techniques in set-theoretic topology

Tall, Franklin D.

doi:10.1090/conm/533/10508

Cited by 3 publications

(3 citation statements)

References 27 publications

(42 reference statements)

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“…We can then transfer a filter G which is Lv(κ, ω 1 ) generic over V to get a filter G * which is j(Lv(κ, ω 1 )) generic over M such that j(p) ∈ G * whenever p ∈ G. This allows us to extend j to an elementary embedding from V [G] to M [G * ]. For an example of this technique, see the proof of Theorem 4.16 in [18].…”

Section: Applications Of X/mmentioning

confidence: 99%

A quotient-like construction involving elementary submodels

Burton

2015

Preprint

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This article is an investigation of a method of deriving a topology from a space and an elementary submodel containing it. We first define and give the basic properties of this construction, known as X/M . In the next section, we construct some examples and analyse the topological relationship between X and X/M . In the final section, we apply X/M to get novel results about Lindelöf spaces, giving partial answers to a question of F.D. Tall and another question of Tall and M. Scheepers.1 Definitions and preliminaries.We assume all spaces are T 3 1 2 -since we are primarily interested in strengthenings of the Lindelöf property, this causes no serious loss of generality. Given an elementary submodel M ≺ H λ and a space X with X ∈ M , we will refer to the classical notion of 'subspace with respect to M ' as X M . Recall that this is the topology on X ∩ M generated by the family (U ∩ M : U ∈ M is open in X) [7]. Our object of study is X/M , which can be seen as analogous to X M where 'subspace' is replaced by 'quotient'. The construction of X/M was introduced independently by I. Bandlow in [1] and A. Dow in [2], and discussed further by T. Eisworth in [3]. There are two related ways to define X/M ; we give the more elementary method first.Definition 1.1. Define an equivalence relation on X by letting x 0 ∼ x 1 if and only if f (x 0 ) = f (x 1 ) for all continuous f : X → R such that f ∈ M . Let X/M be the resulting quotient set and write π : X ։ X/M for the projection. We topologize X/M by taking a base to be all sets of the form π(U ), where U ∈ M is a cozero set in X.Definition 1.2. Since X is T 3 1 2 , we can consider the natural embedding e : X ֒→ [0, 1] C * (X) given by e(x) = (f (x)) f ∈C * (X) . There is also a natural mapThe equivalence of these formulations is proven in [3], with the correspondence given by sending the equivalence class [x] ∈ X/M of a point in x ∈ X to F • e(x). We will use them interchangably. We now summarize the relevant basic properties of X/M .

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Section: Applications Of X/mmentioning

confidence: 99%

A quotient-like construction involving elementary submodels

Burton

2015

Preprint

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“…We can also consider preservation of Lindelöf and D by other kinds of forcing. For example, it is known that a space is Lindelöf in a Cohen or random real extension if and only if it is in the ground model [12], [17], [35], and [39]. The situation for D is more complicated; in [5] it is shown that a Lindelöf space X becomes a D-space in an extension by more than |X| Cohen reals.…”

Section: Other Forcingsmentioning

confidence: 99%

Lindelöf spaces which are indestructible, productive, or D

Aurichi¹,

Tall²

2012

Topology and its Applications

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“…Consolidados como ferramenta, não tardou para que os submodelos elementares passassem a ser tratados também como objeto de estudo. Segundo [1], o primeiro estudo sistemático da técnica em si foi feito em [3]. Lá, dado um espaço topológico X, τ em um submodelo elementar M , X M é o espaço X ∩ M munido da topologia τ M que tem como base a família { U ∩ M : U ∈ τ ∩ M }.…”

Section: Introductionunclassified

Preservação da propriedade de Lindelöf e de algumas de suas generalizações

Figueiredo¹

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Neste trabalho são estudadas a preservação de algumas propriedades sob duas operações: a de tomar um espaço e considerar o seu G δ -refinamento e aquela de tomar um espaço X e um submodelo elementar M e considerar uma certa "versão M de X". Provamos que para espaços dispersos, algumas propriedades como a metacompacidade, a paralindeloficidade e a metalindeloficidade são preservadas por G δ -refinamentos. Também na classe dos espaços dispersos, temos a preservação por submodelos elementares das propriedades de Lindelöf, paracompacidade e metacompacidade. Por fim, apresentamos algumas caracterizações para os D-espaços paracompactos.

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Some problems and techniques in set-theoretic topology

Abstract: Abstract. I survey some problems and techniques that have interested me over the years, e.g. normality vs. collectionwise normality, reflection, preservation by forcing, forcing with Souslin trees, and Lindelöf problems.

Cited by 3 publications

References 27 publications

A quotient-like construction involving elementary submodels

A quotient-like construction involving elementary submodels

Lindelöf spaces which are indestructible, productive, or D

Preservação da propriedade de Lindelöf e de algumas de suas generalizações

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