A countable Fréchet–Urysohn space of uncountable character

Šimon, Petr

doi:10.1016/j.topol.2008.02.001

Cited by 5 publications

(9 citation statements)

References 8 publications

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“…We recall that an AD-family is said to be completely separable if for every M ∈ B * , there is B ∈ B such that B ⊆ M . In the paper [9], P. Simon showed, in ZF C, the existence of a completely separable N M AD-family of size c. Proof. Let A be a completely separable N M AD-family of size c. In the article [9], the author showed that this family satisfies that |{A ∈ A : |M ∩ A| = ω}| = c for all M ∈ A * .…”

Section: Let Us Remark That Ifmentioning

confidence: 99%

“…In the paper [9], P. Simon showed, in ZF C, the existence of a completely separable N M AD-family of size c. Proof. Let A be a completely separable N M AD-family of size c. In the article [9], the author showed that this family satisfies that |{A ∈ A : |M ∩ A| = ω}| = c for all M ∈ A * . If M ∈ S + A and |{A ∈ A : |M ∩ A| = ω}| < ω, then S A | M is a relatively equivalent to the Fréchet filter.…”

Section: Let Us Remark That Ifmentioning

confidence: 99%

“…Next, let us describe another useful construction of F U -filters which has been very important in the construction of special F U -filters (see, for instance, the article [9]):…”

Section: Fréchet-urysohn Filtersmentioning

confidence: 99%

“…Theorem 2.3. ( [9]) Given an AD-family A, we have that the filter S A is a F Ufilter iff A is a N M AD-family.…”

Section: Fréchet-urysohn Filtersmentioning

confidence: 99%

“…The F AN -filter is a canonical example of a F U -filter which is not an α 4 -filter; indeed, it is well-know that a space is not an α 4 -space iff the space contains a copy of F AN -space (for a prove see [10]). In the article [9], P. Simon constructed a completely separable N M AD-family A of size c such that S A is an α 4 -filter which is not an α 3 -filter. For this AD-family A, it is easy to show that F A is also an α 4 -filter that is not an α 3 -filter.…”

Section: Rk-incomparability Of F U -Filtersmentioning

confidence: 99%

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Comparing Fréchet–Urysohn filters with two pre-orders

Garcı́a-Ferreira

Rivera-Gómez

2017

Topology and its Applications

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Abstract. A filter F on ω is called Fréchet-Urysohn if the space with only one non-isolated point ω∪{F } is a Fréchet-Urysohn space, where the neighborhoods of the non-isolated point are determined by the elements of F . In this paper, we distinguish some Fréchet-Urysohn filters by using two pre-orderings of filters: One is the Rudin-Keisler pre-order and the other one was introduced by Todorčević-Uzcátegui in [11]. In this paper, we construct an RK-chain of size c + which is RK-above of avery F U -filter. Also, we show that there is an infinite RK-antichain of F U -filters.

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Section: Let Us Remark That Ifmentioning

confidence: 99%

Section: Let Us Remark That Ifmentioning

confidence: 99%

“…Next, let us describe another useful construction of F U -filters which has been very important in the construction of special F U -filters (see, for instance, the article [9]):…”

Section: Fréchet-urysohn Filtersmentioning

confidence: 99%

“…Theorem 2.3. ( [9]) Given an AD-family A, we have that the filter S A is a F Ufilter iff A is a N M AD-family.…”

Section: Fréchet-urysohn Filtersmentioning

confidence: 99%

Section: Rk-incomparability Of F U -Filtersmentioning

confidence: 99%

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Comparing Fréchet–Urysohn filters with two pre-orders

Garcı́a-Ferreira

Rivera-Gómez

2017

Topology and its Applications

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Workshop lecture on products of Fréchet spaces

Nyikos

2010

Topology and its Applications

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The general question, "When is the product of Fréchet spaces Fréchet?" really depends on the questions of when a product of α 4 Fréchet spaces (also known as strongly Fréchet or countably bisequential spaces) is α 4 , and when it is Fréchet. Two subclasses of the class of strongly Fréchet spaces shed much light on these questions. These are the class of α 3 Fréchet spaces and its subclass of ℵ 0-bisequential spaces. The latter is closed under countable products, the former not even under finite products. A number of fundamental results and open problems are recalled, some further highlighting the difference between being α 3 and Fréchet and being ℵ 0-bisequential. This paper is a slightly updated note for the first of two lectures presented by the author in his Workshop on Sequential Convergence at the ten-day Advances in Set-Theoretic Topology conference in Erice. Erice is a remarkable mountaintop town with a medieval feel to it, overlooking the northwestern tip of Sicily, and the author is grateful for the invitation to this unique conference. Recall that a space is called Fréchet (or: Fréchet-Urysohn) if, whenever a point p is in the closure of a subset A, there is a sequence in A converging to p. In this paper, "space" will mean "Hausdorff space," although much of what we say holds for topological spaces in general. This paper revolves around the following general problem, to which Tsugunori Nogura has made many basic contributions. General Problem. When is the product of Fréchet spaces Fréchet? 1. Fundamental problems and theorems The following space is very relevant to this general problem.

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