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On the cardinality and weight spectra of compact spaces, II

100%

Juhász I., Shelah S.

Fundamenta Mathematicae

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1998

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tom 155

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nr 1

91-94

EN

Let B(κ,λ) be the subalgebra of P(κ) generated by $[κ]^{≤λ}$. It is shown that if B is any homomorphic image of B(κ,λ) then either $|B| < 2^λ$ or $|B| = |B|^λ$; moreover, if X is the Stone space of B then either $|X| ≤ 2^{2^λ}$ or $|X| = |B| = |B|^λ$. This implies the existence of 0-dimensional compact $T_2$ spaces whose cardinality and weight spectra omit lots of singular cardinals of "small" cofinality.

2

Are initially $ω_1$ -compact separable regular spaces compact?

100%

Dow A., Juhász I.

Fundamenta Mathematicae

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1997

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tom 154

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nr 2

123-132

EN

We investigate the question of the title. While it is immediate that CH yields a positive answer we discover that the situation under the negation of CH holds some surprises.

3

Strongly almost disjoint familes, revisited

81%

Hajnal A., Juhász I., Shelah S.

Fundamenta Mathematicae

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2000

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tom 163

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nr 1

13-23

EN

The relations M(κ,λ,μ) → B [resp. B(σ)] meaning that if $A⊂[κ]^λ$ with |A|=κ is μ-almost disjoint then A has property B [resp. has a σ-transversal] had been introduced and studied under GCH in [EH]. Our two main results here say the following: Assume GCH and let ϱ be any regular cardinal with a supercompact [resp. 2-huge] cardinal above ϱ. Then there is a ϱ-closed forcing P such that, in $V^P$, we have both GCH and $M(ϱ^{(+ϱ+1)},ϱ^+,ϱ) ↛ B$ [resp. $M(ϱ^{(+ϱ+1)},λ,ϱ) ↛ B(ϱ^+)$ for all $λ ≤ ϱ^{(+ϱ+1)}]$. These show that, consistently, the results of [EH] are sharp. The necessity of using large cardinals follows from the results of [Ko], [HJSh] and [BDJShSz].

Ograniczanie wyników

3 Fundamenta Mathematicae

3 Juhász I.

2 Shelah S.

1 Dow A.

1 Hajnal A.

1 2000

1 1998

1 1997

On the cardinality and weight spectra of compact spaces, II

Are initially $ω_1$ -compact separable regular spaces compact?

Strongly almost disjoint familes, revisited