Some Applications of Almost Disjoint Sets

Jensen, Ronald Björn; Solovay, Robert M.

doi:10.1016/s0049-237x(08)71932-3

Cited by 63 publications

(62 citation statements)

References 2 publications

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“…This is the form used most often in set theory [7], and it is commonly used in topology as well (see, e.g., [23]). For example, it is essentially shown in [4] that q = ℵ 1 implies that there is an uncountable E ⊂ [0, 1] such that the square of S(E) is T 5 .…”

Section: Lemma 61 ([22] See Also [3]) a Compact Space Is Metrizablmentioning

confidence: 99%

Katetov’s problem

Larson

Todorčević²

2001

Trans. Amer. Math. Soc.

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Section: Lemma 61 ([22] See Also [3]) a Compact Space Is Metrizablmentioning

confidence: 99%

Katetov’s problem

Larson

Todorčević²

2001

Trans. Amer. Math. Soc.

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“…By using almost disjoint coding (see [JS70] and [Har77, Section 1]), the above assumption can be seen to hold in every model of…”

Section: ])mentioning

confidence: 99%

Continuous images of closed sets in generalized Baire spaces

Lücke

Schlicht

2015

Isr. J. Math.

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Abstract. Let κ be an uncountable cardinal with κ = κ <κ . Given a cardinal µ, we equip the set κ µ consisting of all functions from κ to µ with the topology whose basic open sets consist of all extensions of partial functions of cardinality less than κ. We prove results that allow us to separate several classes of subsets of κ κ that consist of continuous images of closed subsets of spaces of the form κ µ. Important examples of such results are the following: (i) there is a closed subset of κ κ that is not a continuous image of κ κ; (ii) there is an injective continuous image of κ κ that is not κ-Borel (i.e. that is not contained in the smallest algebra of sets on κ κ that contains all open subsets and is closed under κ-unions); (iii) the statement " every continuous image of κ κ is an injective continuous image of a closed subset of κ κ " is independent of the axioms of ZFC; and (iv) the axioms of ZFC do not prove that the assumption " 2 κ > κ + " implies the statement " every closed subset of κ κ is a continuous image of κ (κ + ) " or its negation.

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“…Now we must code F by a subset of co,. The standard way to do this is with almost disjoint set forcing, invented in Jensen and Solovay [1970] and perfected in Jensen [1972,1975]. This method can be described as follows: First choose a A,(LJF], B) sequence (X | y < a) of subsets of co, such that y ¥= y' -» X fl X , is countable (Xy, Xy, are almost disjoint).…”

Section: Killing Admissibles Our Goal In This Section Is To Establishmentioning

confidence: 99%

Uncountable admissibles. I. Forcing

Friedman¹

1982

Trans. Amer. Math. Soc.

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Abstract. Assume V = L. Let k be a regular cardinal and for X Qk let a(X) denote the least ordinal a such that ¿"[A"] is admissible. In this paper we characterize those ordinals of the form k s.t. La( X) is admissible.Question (V -L). Which admissible ordinals are of the form a(X) for some X QkI We deal in this paper with the case where k is regular. (The singular case will be treated in Friedman [1981a].) The answer to this Question is best phrased in terms of a strong form of < «-closure which we call < K-admissibility.

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Some Applications of Almost Disjoint Sets

Cited by 63 publications

References 2 publications

Katetov’s problem

Katetov’s problem

Continuous images of closed sets in generalized Baire spaces

Uncountable admissibles. I. Forcing

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