Left separated spaces with point-countable bases

Fleissner, William G.

doi:10.1090/s0002-9947-1986-0825729-x

Cited by 25 publications

(21 citation statements)

References 14 publications

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“…This principle proved to be a very useful infinite-combinatorial tool for proving many statements originally known to be consequences of Fleissner's Axiom R ( [5]) 1 . Then, by studying the set-theoretic characterizations of FRP, we found out that the principle is even equivalent to many of these statements and the list of the mathematical statements equivalent to FRP over ZFC is growing ever since.…”

Section: )mentioning

confidence: 99%

A Reflection Principle As a Reverse-mathematical Fixed Point over the Base Theory ZFC

Fuchino

2017

Annals of the Japan Association for Philosophy of Science

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We survey the study of the Fodor-type Reflection Principle (FRP) and discuss the significance of the principle in terms of what we call the reverse-mathematical criterion.Key words: Fodor-type Reflection Principle (FRP), Shelah's Strong Hypothesis (SSH), reverse-mathematical criterion Reverse-mathematical fixed pointsIn reverse mathematics, five subtheories of second order arithmetic (the so called Big 5) are put under the spotlight. These theories are considered as being central, for one thing, because each of them is equivalent to many classical theorems over the base theory RCA 0 . For example the axiom system ACA 0 which is often considered to be the modern representation of the system Hermann Weyl introduced in [39] (see also [17]) is known to be equivalent to the system RCA 0 + Bolzano-Weierstrass theorem or to the system RCA 0 + Ascoli's theorem (see e.g. [33] for more such examples).The same criterion for determining the significance (or at least the relevance) of axioms (or principles) can be also applied to assertions over other base theories.The Axiom of Choice (AC) for example can be considered to be significant because of a long list of mathematical theorems which are known to be equivalent to it over the base theory ZF. Some of such theorems are:

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Section: )mentioning

confidence: 99%

A Reflection Principle As a Reverse-mathematical Fixed Point over the Base Theory ZFC

Fuchino

2017

Annals of the Japan Association for Philosophy of Science

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“…Under Fleissner's Axiom R [7], there is no need to consider cardinals higher than ω 1 for the topological application of Section 5 (see Theorem 23).…”

Section: Proof ♦(S) Implies That There Are Fmentioning

confidence: 99%

“…Theorem 25 ( [7]). (Axiom R) If X is ω 1 -cwH , has local density ≤ ω 1 , and has countable tightness, then X is CW H. Finally, we mention that we also do not know if thin + M 0 is consistent, i.e., if there could be a thin, normal ladder system.…”

Section: Proof ♦(S) Implies That There Are Fmentioning

confidence: 99%

Uniformization and anti-uniformization properties of ladder systems

et al. 2004

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Abstract. Natural weakenings of uniformizability of a ladder system on ω 1 are considered. It is shown that even assuming CH all the properties may be distinct in a strong sense. In addition, these properties are studied in conjunction with other properties inconsistent with full uniformizability, which we call anti-uniformization properties. The most important conjunction considered is the uniformization property we call countable metacompactness and the anti-uniformization property we call thinness. The existence of a thin, countably metacompact ladder system is used to construct interesting topological spaces: a countably paracompact and nonnormal subspace of ω 2 1 , and a countably paracompact, locally compact screenable space which is not paracompact. Whether the existence of a thin, countably metacompact ladder system is consistent is left open. Finally, the relation between the properties introduced and other well known properties of ladder systems, such as ♣, is considered.

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“…Em todo caso, somente escreveremos uma fórmula em português quando for claro que a mesma pode ser expressa com símbolos da linguagem de teoria dos conjuntos. 11 Ou seja, é do tipo "x 1 = x 2 " ou "x 1 ∈ x 2 ".…”

Section: Submodelos Elementaresunclassified

“…Como (Z, τ ′ ) é T 2 e tem caráter enumerável, o mesmo ocorre com (X, τ ); além disso, (X, τ ) 9 Muitas vezes, substituindo-se "coletivamente de Hausdorff" por "λ-coletivamente de Hausdorff" e "ω 1coletivamente de Hausdorff" por "κ-coletivamente de Hausdorff para todo cardinal κ < λ", sendo κ um cardinal maior ou igual a ω 2 -vide, e.g., [9] e [30]. 10 Cabe citar que a pergunta de reflexão também é estudada; vide, e.g., [11].…”

Section: Espaços ω 1 -Coletivamente De Hausdorffunclassified

Reflexão de funções cardinais e da metrizabilidade

Dias¹

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O conceito de reflexão em topologia expressa o fato de que um espaço satisfaz uma dada propriedade sempre que esta é satisfeita por seus subespaços "menores". Neste trabalho, estuda-se a reflexão de propriedades envolvendo a maioria das principais funções cardinais e metrizabilidade, bem como outras propriedades relacionadas. São discutidos problemas em aberto-como o problema de Hamburger-, incluindo respostas parciais e exemplos de consistência. Várias dentre as demonstrações apresentadas utilizam técnicas de submodelos elementares, que constituem hoje uma importante ferramenta no estudo de topologia geral.

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Left separated spaces with point-countable bases

Cited by 25 publications

References 14 publications

A Reflection Principle As a Reverse-mathematical Fixed Point over the Base Theory ZFC

A Reflection Principle As a Reverse-mathematical Fixed Point over the Base Theory ZFC

Uniformization and anti-uniformization properties of ladder systems

Reflexão de funções cardinais e da metrizabilidade

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