Continuing horrors of topology without choice

Good, Chris; Tree, I.J.

doi:10.1016/0166-8641(95)90010-1

Cited by 76 publications

(39 citation statements)

References 8 publications

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“…The fact that every separable metrizable space is paracompact can be proved from ZF [4]. That every second countable metric space is paracompact, Dieudonné's original result, can also be so proved (but recall from [4] that there are models of ZF containing second countable metric spaces that are not separable).…”

Section: Introductionmentioning

confidence: 83%

“…That every second countable metric space is paracompact, Dieudonné's original result, can also be so proved (but recall from [4] that there are models of ZF containing second countable metric spaces that are not separable). However, we show that this is not true of the general theorem: we construct symmetric models of ZF in which there are metric spaces that are not paracompact.…”

Section: Introductionmentioning

confidence: 90%

“…So, as we see, even the most innocent of topological questions may be undecidable from the ZermeloFraenkel axioms alone. Further examples of the counter-intuitive behaviour of 'choiceless topology' can be found in [3] and [4].…”

Section: Introductionmentioning

confidence: 99%

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On Stone’s theorem and the Axiom of Choice

Good¹,

Tree²,

Watson

1998

Proc. Amer. Math. Soc.

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Abstract. It is a well established fact that in Zermelo-Fraenkel set theory, Tychonoff's Theorem, the statement that the product of compact topological spaces is compact, is equivalent to the Axiom of Choice. On the other hand, Urysohn's Metrization Theorem, that every regular second countable space is metrizable, is provable from just the ZF axioms alone. A. H. Stone's Theorem, that every metric space is paracompact, is considered here from this perspective. Stone's Theorem is shown not to be a theorem in ZF by a forcing argument. The construction also shows that Stone's Theorem cannot be proved by additionally assuming the Principle of Dependent Choice.

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Section: Introductionmentioning

confidence: 83%

Section: Introductionmentioning

confidence: 90%

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On Stone’s theorem and the Axiom of Choice

Good¹,

Tree²,

Watson

1998

Proc. Amer. Math. Soc.

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“…C. Good and I. Tree [2] establish that the latter statements imply CAC fin (the axiom of choice for countable families of non empty finite sets). Further, H. L. Bentley and H. Herrlich [1] consider the pseudometric versions of the above statements, i. e.…”

Section: Introduction and Preliminary Resultsmentioning

confidence: 92%

On Lindelöf Metric Spaces and Weak Forms of the Axiom of Choice

Keremedis

Tachtsis

2000

MLQ - Math. Log. Quart.

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“…As an example, AC and Tychonoff's compactness theorem express the same truth in mathematics (see [10]). For horrors and disasters which we may encounter in topology without AC, we refer the reader to [1], [3], [4], [5], [7], [11], [14], and [16].…”

Section: Introduction and Some Preliminary Resultsmentioning

confidence: 99%