On Sequential Compactness and Related Notions of Compactness of Metric Spaces in $\mathbf {ZF}$

Keremedis, Kyriakos

doi:10.4064/ba8054-5-2016

Cited by 11 publications

(20 citation statements)

References 7 publications

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“…In view of (i), it is obvious that (ii) holds. It is known from [23] that (iii) also holds. It follows from the first implication of (i) that to prove (iv), it suffices to show that M(T B, W O) does not imply CAC.…”

Section: Theorem 9 (Zf) (I) M(t B W O) → M(t B S) and M(c W O) → M(c S) None Of These Implications Are Reversible (Ii) M(t B W O) → M(c Wmentioning

confidence: 90%

“…For every totally bounded metric space X , d , the set X is wellorderable. The notation of type M(C, ) was introduced in [22] and was also used in [23], but not all forms from the definition above were defined in [22,23]…”

Section: M(t B W O)mentioning

confidence: 99%

“…It was shown in [23,proof of Theorem 15] that there exists a model M of ZF + ¬CAC in which it is true that if a metric space X = X , d is sequentially bounded (i.e., every sequence of points of X has a Cauchy's subsequence), then X is wellorderable and separable. By [23, Theorem 7 (vii)], every totally bounded metric space is sequentially bounded.…”

Section: Theorem 9 (Zf) (I) M(t B W O) → M(t B S) and M(c W O) → M(c S) None Of These Implications Are Reversible (Ii) M(t B W O) → M(c Wmentioning

confidence: 99%

“…We recall that a topological space X is called limit point compact if every infinite subset of X has an accumulation point in X (see, e.g., [23]).…”

Section: (Ii) In Cohen's Original Model M1 Of [15] the Following Hold: M(c S) Is True But Both M(t B S) And M(t B St B) Are False (Iii) Mmentioning

confidence: 99%

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

2021

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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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Section: Theorem 9 (Zf) (I) M(t B W O) → M(t B S) and M(c W O) → M(c S) None Of These Implications Are Reversible (Ii) M(t B W O) → M(c Wmentioning

confidence: 90%

Section: M(t B W O)mentioning

confidence: 99%

Section: Theorem 9 (Zf) (I) M(t B W O) → M(t B S) and M(c W O) → M(c S) None Of These Implications Are Reversible (Ii) M(t B W O) → M(c Wmentioning

confidence: 99%

“…We recall that a topological space X is called limit point compact if every infinite subset of X has an accumulation point in X (see, e.g., [23]).…”

Section: (Ii) In Cohen's Original Model M1 Of [15] the Following Hold: M(c S) Is True But Both M(t B S) And M(t B St B) Are False (Iii) Mmentioning

confidence: 99%

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

2021

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“…(d) [9] (ZF) X is compact iff it is totally bounded and Lebesgue. (e) [7] (ZF) Every sequentially compact metric space is compact iff every compact metric space is separable. Table 2 records the implications/non-implications which hold between the forms of compactness listed in Theorem 1.…”

mentioning

confidence: 99%

Some Notions of Separability of Metric Spaces in $\mathbf {ZF}$ and Their Relation to Compactness

Keremedis

2016

Bull. Polish Acad. Sci. Math.

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In the realm of metric spaces we show in ZF that: (1) Quasi separability (a metric space X = (X, d) is quasi separable iff X has a dense subset which is expressible as a countable union of finite sets) is the weakest property under which a limit point compact metric space is compact. (2) ω-quasi separability (a metric space X = (X, d) is ω-quasi separable iff X has a dense subset which is expressible as a countable union of countable sets) is a property under which a countably compact metric space is compact. (3) The statement "Every totally bounded metric space is separable" does not imply the countable choice axiom CAC.

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