On Loeb and sequential spaces in ZF

In the absence of the axiom of choice, the set-theoretic status of many natural statements about metrizable compact spaces is investigated. Some of the statements are provable in $$\mathbf {ZF}$$ ZF , some are shown to be independent of $$\mathbf {ZF}$$ ZF . For independence results, distinct models of $$\mathbf {ZF}$$ ZF and permutation models of $$\mathbf {ZFA}$$ ZFA with transfer theorems of Pincus are applied. New symmetric models of $$\mathbf {ZF}$$ ZF are constructed in each of which the power set of $$\mathbb {R}$$ R is well-orderable, the Continuum Hypothesis is satisfied but a denumerable family of non-empty finite sets can fail to have a choice function, and a compact metrizable space need not be embeddable into the Tychonoff cube $$[0, 1]^{\mathbb {R}}$$ [ 0 , 1 ] R .

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“…(ii) (Cf. [30].) If X is a compact second-countable and metrizable space, then X ω is compact and separable.…”

Section: Theorem 6 (Zf)mentioning

confidence: 99%

Several results on compact metrizable spaces in $$\mathbf {ZF}$$

2021

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“…(ii) (Cf. [25].) If X is a compact second-countable and metrizable space, then X ω is compact and separable.…”

Section: A List Of Several Known Theoremsmentioning

confidence: 99%

Countable products and countable sums of compact metrizable spaces in the absence of the Axiom of Choice

Keremedis,

Tachtsis,

Wajch

2021

Preprint

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The main aim of the article is to show, in the absence of the Axiom of Choice, relationships between the following, independent of ZF, statements: "Every countable product of compact metrizable spaces is separable (respectively, compact)" and "Every countable product of compact metrizable spaces is metrizable". Statements related to the above-mentioned ones are also studied. Permutation models (among them new ones) are shown in which a countable sum (also a countable product) of metrizable spaces need not be metrizable, countable unions 1 of countable sets are countable and there is a countable family of nonempty sets of size at most 2 ℵ 0 which does not have a choice function. A new permutation model is constructed in which every uncountable compact metrizable space is of size at least 2 ℵ 0 but a denumerable family of denumerable sets need not have a multiple choice function.

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“…Basic facts about sequentially closed sets and sequential spaces in ZF can be found, for instance, in [6, Section 4.6], [10], [16] and [17].…”

Section: Some Known Theoremsmentioning

confidence: 99%

$k$-spaces, sequential spaces and related topics in the absence of the axiom of choice

Keremedis¹,

Wajch²

2021

Preprint

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In the absence of the axiom of choice, new results concerning sequential, Fréchet-Urysohn, k-spaces, very k-spaces, Loeb and Cantor completely metrizable spaces are shown. New choice principles are introduced. Among many other theorems, it is proved in ZF that every Loeb, T 3 -space having a base expressible as a countable union of finite sets is a metrizable second-countable space whose every F σ -subspace is separable; moreover, every G δ -subspace of a second-countable, Cantor completely metrizable space is Cantor completely metrizable, Loeb and separable. It is also noticed that Arkhangel'skii's statement that every very k-space is Fréchet-Urysohn is unprovable in ZF but it holds in ZF that every first-countable, regular very k-space whose family of all nonempty compact sets has a choice function is Fréchet-Urysohn. That every second-countable metrizable space is a very k-space is equivalent to the axiom of countable choice for R.

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On Loeb and sequential spaces in ZF

Cited by 9 publications

References 16 publications

Several results on compact metrizable spaces in $$\mathbf {ZF}$$

Several results on compact metrizable spaces in $$\mathbf {ZF}$$

Countable products and countable sums of compact metrizable spaces in the absence of the Axiom of Choice

$k$-spaces, sequential spaces and related topics in the absence of the axiom of choice

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