Hausdorff compactifications in ZF

Keremedis, Kyriakos; Wajch, Eliza

doi:10.1016/j.topol.2019.02.046

Cited by 9 publications

(21 citation statements)

References 18 publications

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“…(Cf. [23].) (ZF) (i) For every non-empty compact Hausdorff space K, there exists a Dedekind-infinite discrete space D such that K is a remainder of D. Then: [25]);…”

Section: Some Known Resultsmentioning

confidence: 99%

“…(i) and[23, Theorem 3.27], it follows from BPI that there exists the Čech-Stone compactification βD of D. Suppose that βD \ D has an isolated point y 0 . Then there exist disjoint open subsets U, V of βD such that y 0 ∈ U, (βD \ D) \ {y 0 } ⊆ V and U ∩ V = ∅.…”

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confidence: 98%

“…Therefore, every one-point Hausdorff compactification of X is called the Alexandroff compactification of X. Chandler's book [3] is a good introduction to Hausdorff compactifications in ZFC. Basic facts about Hausdorff compactifications in ZF can be found in [23]. If X is a space which has the Čech-Stone compactification, then, as usual, βX stands for the Čech-Stone compactification of X.…”

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confidence: 99%

See 2 more Smart Citations

On iso-dense and scattered spaces in $\mathbf{ZF}$

Keremedis¹,

Tachtsis²,

Wajch³

2021

Preprint

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A topological space is iso-dense if it has a dense set of isolated points. A topological space is scattered if each of its non-empty subspaces has an isolated point. In ZF, in the absence of the axiom of choice, basic properties of iso-dense spaces are investigated. A new permutation model is constructed in which a discrete weakly Dedekindfinite space can have the Cantor set as a remainder. A metrization theorem for a class of quasi-metric spaces is deduced. The statement "every compact scattered metrizable space is separable" and several other statements about metric iso-dense spaces are shown to be equivalent to the countable axiom of choice for families of finite sets. Results concerning the problem of whether it is provable in ZF that every non-discrete compact metrizable space contains an infinite compact scattered subspace are also included.

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“…(Cf. [23].) (ZF) (i) For every non-empty compact Hausdorff space K, there exists a Dedekind-infinite discrete space D such that K is a remainder of D. Then: [25]);…”

Section: Some Known Resultsmentioning

confidence: 99%

mentioning

confidence: 98%

mentioning

confidence: 99%

See 1 more Smart Citation

On iso-dense and scattered spaces in $\mathbf{ZF}$

Keremedis¹,

Tachtsis²,

Wajch³

2021

Preprint

Self Cite

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“…In ZFC, every Tychonoff space has its Čech-Stone compactification; however, in a model of ZF, a Tychonoff space may fail to have its Čech-Stone compactification (cf., e.g., [22,Theorem 3.7]). The book [2] is a very good introduction to Hausdorff compactifications in ZFC.…”

Section: The Collectionmentioning

confidence: 99%

Second-countable compact Hausdorff spaces as remainders in $\mathbf{ZF}$ and two new notions of infiniteness

Keremedis,

Tachtsis,

Wajch

2020

Preprint

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In the absence of the Axiom of Choice, necessary and sufficient conditions for a locally compact Hausdorff space to have all non-empty second-countable compact Hausdorff spaces as remainders are given in ZF. Among other independence results, the characterization of locally compact Hausdorff spaces having all non-empty metrizable compact spaces as remainders, obtained by Hatzenhuhler and Mattson in ZFC, is proved to be independent of ZF. Urysohn's Metrization Theorem is generalized to the following theorem: every T 3 -space which admits a base expressible as a countable union of finite sets is metrizable. Applications to solutions of problems concerning the existence of some special metrizable compactifications in ZF are shown. New concepts of a strongly filterbase infinite set and a dyadically filterbase infinite set are introduced, both stemming from the investigations on compactifications. Set-theoretic and topological definitions of the new concepts are given, and their relationship with certain known notions of infinite sets is investigated in ZF. A new permutation model is introduced in which there exists a strongly filterbase infinite set which is weakly Dedekind-finite. All ZFA-independence results of this article are transferable to ZF.

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“…It has been shown, for instance, in [6] and [7] recently that generalized topologies that are not topologies can appear in a very natural way in some mathematical problems. Needless to say, the set-theoretic strength of Urysohn's Lemma and the Tietze-Urysohn Extension Theorem for topological spaces is very important (see, e.g., [8,Forms 78 and 375], [9] and [11]). In [3], a modification of Urysohn's Lemma for normal generalized topological spaces was obtained.…”

Section: Introductionmentioning

confidence: 99%

On Urysohn's Lemma for generalized topological spaces in ZF

Hejduk¹,

Wajch²

2021

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A strong generalized topological space is an ordered pair X = X, T such that X is a set and T is a collection of subsets of X such that ∅, X ∈ T and T is stable under arbitrary unions. A necessary and sufficient condition for a strong generalized topological space X to satisfy Urysohn's lemma or its appropriate variant is shown in ZF. Notions of a U-normal and an effectively normal generalized topological space are introduced. It is observed that, in ZF + DC, every U-normal generalized topological space satisfies Urysohn's lemma. It is shown that every effectively normal generalized topological space satisfies Csaszár's modification of Urysohn's Lemma. A ZF-example of a strong generalized topological normal space which satisfies the Tietze-Urysohn Extension Theorem and fails to satisfy Urysohn's Lemma is shown.

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Hausdorff compactifications in ZF

Cited by 9 publications

References 18 publications

On iso-dense and scattered spaces in $\mathbf{ZF}$

On iso-dense and scattered spaces in $\mathbf{ZF}$

Second-countable compact Hausdorff spaces as remainders in $\mathbf{ZF}$ and two new notions of infiniteness

On Urysohn's Lemma for generalized topological spaces in ZF

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