Additions to some results of Erdös and Tarski

Monk, Donald; Scott, Dana

doi:10.4064/fm-53-3-335-343

Cited by 21 publications

(9 citation statements)

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“…We shall say that a topological space X is κ ‐compact (or κ‐Lindelöf) if every open cover of X has a subcover of cardinality less than κ (cf. [18, 36]). Obviously, the Cantor space 2 ω is ω‐compact (i.e., compact in the traditional sense).…”

Section: Introductionmentioning

confidence: 99%

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Special subsets of the generalized Cantor space and generalized Baire space

Korch

Weiss

2020

Mathematical Logic Qtrly

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In this paper, we are interested in parallels to the classical notions of special subsets in R defined in the generalized Cantor and Baire spaces (2 κ and κ κ). We consider generalizations of the well-known classes of special subsets, like Lusin sets, strongly null sets, concentrated sets, perfectly meagre sets, σ-sets, γ-sets, sets with the Menger, the Rothberger, or the Hurewicz property, but also of some less-know classes like X-small sets, meagre additive sets, Ramsey null sets, Marczewski, Silver, Miller, and Laver-null sets. We notice that many classical theorems regarding these classes can be relatively easy generalized to higher cardinals although sometimes with some additional assumptions. This paper serves as a catalogue of such results along with some other generalizations and open problems.

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Section: Introductionmentioning

confidence: 99%

“…Actually, the κ‐Cantor space 2 κ is κ‐compact if and only if κ is a weakly compact cardinal (cf. [36]). And there is even more to that: The κ‐Cantor space 2 κ and the κ‐Baire space

κ^{κ}

are homeomorphic if and only if κ is not a weakly compact cardinal (cf.…”

Section: Introductionmentioning

confidence: 99%

Special subsets of the generalized Cantor space and generalized Baire space

Korch

Weiss

2020

Mathematical Logic Qtrly

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“…The non-existence of κ-Aronszajn trees (the tree property at κ) and the nonexistence of special κ-Aronszajn trees (failure of * ) are reflection principles which are closely connected with large cardinals. For example, theorems of Erdös and Tarski [6], and Monk and Scott [14], show that an inaccessible cardinal is weakly compact if and only if it has the tree property. Further, Mitchell and Silver [13] showed that the tree property at ℵ 2 is consistent with ZFC if and only if the existence of a weakly compact cardinal is.…”

Section: Introductionmentioning

confidence: 99%

Diagonal supercompact Radin forcing

Ben-Neria¹,

Lambie-Hanson²,

Unger³

2016

Preprint

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“…Various other characterizations of weakly compact cardinals are known (cf. [15], [21], [25], [27]); these are not used in this paper. A weakly compact cardinal is strongly inaccessible.…”

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confidence: 99%

Spaces homeomorphic to (2^{𝑎})ₐ. II

Hung¹,

Negrepontis²

1974

Trans. Amer. Math. Soc.

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Topological characterizations and properties of the spaces (2a)a, where a is an infinite regular cardinal, are studied; the principal interest lying in the significance that these spaces have in questions of existence of ultrafilters (or of elements of the Stone-Cech compactification of spaces) with special properties. The main results are (a) the characterization theorem of the spaces (2a)a in terms of a simple set of conditions, and (b) the a-Baire category property of (2a)a and the stability of the class of spaces homeomorphic to (2a)a (or to (aa)a) when taking intersections of at most a. open and dense sub* sets of (2a)a. Among the applications of these results are the following. Assuming a+ = 2a, the class of spaces homeomorphic to (2(a))a + includes the space of uniform ultrafilters on

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Additions to some results of Erdös and Tarski

Cited by 21 publications

References 0 publications

Special subsets of the generalized Cantor space and generalized Baire space

Special subsets of the generalized Cantor space and generalized Baire space

Diagonal supercompact Radin forcing

Spaces homeomorphic to (2^{𝑎})ₐ. II

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