Continuous extension of maps between sequential cascades

Dolecki, Szymon; Starosolski, Andrzej

doi:10.1016/j.apal.2020.102928

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Theorem 312 (B = C) There Exists An Order-embedding Of the Real Line Into The Set Of P-points1

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2021

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“…(b = c). 4 If p is a P-point, then there exists a P-point q that is strictly Rudin-Keisler greater than p.…”

Section: Quasi-subbasesmentioning

confidence: 99%

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THE RUDIN–KEISLER ORDERING OF P-POINTS UNDER 𝔟 = 𝔠

Starosolski¹

2021

J. symb. log.

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M. E. Rudin (1971) proved, under CH, that for each P-point p there exists a P-point q strictly RK-greater than p. This result was proved under ${\mathfrak {p}= \mathfrak {c}}$ by A. Blass (1973), who also showed that each RK-increasing $ \omega $ -sequence of P-points is upper bounded by a P-point, and that there is an order embedding of the real line into the class of P-points with respect to the RK-ordering. In this paper, the results cited above are proved under the (weaker) assumption that $\mathfrak { b}=\mathfrak {c}$ . A. Blass asked in 1973 which ordinals can be embedded in the set of P-points, and pointed out that such an ordinal cannot be greater than $ \mathfrak {c}^{+}$ . In this paper it is proved, under $\mathfrak {b}=\mathfrak {c}$ , that for each ordinal $\alpha < \mathfrak {c}^{+}$ , there is an order embedding of $ \alpha $ into P-points. It is also proved, under $\mathfrak {b}=\mathfrak {c}$ , that there is an embedding of the long line into P-points.

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“…(b = c). 4 If p is a P-point, then there exists a P-point q that is strictly Rudin-Keisler greater than p.…”

Section: Quasi-subbasesmentioning

confidence: 99%

“…Thus to prove (4), it suffices by Corollary 2.6 to prove that B ∪ C is a P + -family. Thus, by Theorem 2.5, in order to prove (4), it suffices to prove that: D i is a P + -family for D i defined for i < as follows:…”

Section: Theorem 312 (B = C) There Exists An Order-embedding Of the Real Line Into The Set Of P-pointsmentioning

confidence: 99%

THE RUDIN–KEISLER ORDERING OF P-POINTS UNDER 𝔟 = 𝔠

Starosolski¹

2021

J. symb. log.

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Continuous extension of maps between sequential cascades

Cited by 1 publication

References 13 publications

THE RUDIN–KEISLER ORDERING OF P-POINTS UNDER 𝔟 = 𝔠

THE RUDIN–KEISLER ORDERING OF P-POINTS UNDER 𝔟 = 𝔠

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