On what I do not understand (and have something to say): Part I

Saharon Shelah

Displaying similar documents to “On what I do not understand (and have something to say): Part I”

Embedding Cohen algebras using pcf theory

Saharon Shelah (2000)

Fundamenta Mathematicae

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Using a theorem from pcf theory, we show that for any singular cardinal ν, the product of the Cohen forcing notions on κ, κ < ν, adds a generic for the Cohen forcing notion on $ν^{+}$ .

Cellularity of free products of Boolean algebras (or topologies)

Saharon Shelah (2000)

Fundamenta Mathematicae

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The aim this paper is to present an answer to Problem 1 of Monk [10], [11]. We do this by proving in particular that if μ is a strong limit singular cardinal, $θ = {(2^{c f (μ)})}^{+}$ and $2^{μ} = μ^{+}$ then there are Boolean algebras $𝔹_{1}, 𝔹_{2}$ such that $c (𝔹_{1}) = μ, c (𝔹_{2}) < θ b u t c (𝔹_{1} * 𝔹_{2}) = μ^{+}$ . Further we improve this result, deal with the method and the necessity of the assumptions. In particular we prove that if $𝔹$ is a ccc Boolean algebra and $μ^{ℶ_{ω}} \leq λ = c f (λ) \leq 2^{μ}$ then $𝔹$ satisfies the λ-Knaster condition (using the “revised GCH theorem”).

Some variations on the partition property for normal ultrafilters on Pkl

Julius Barbanel (1993)

Fundamenta Mathematicae

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Suppose κ is a supercompact cardinal and λ≥κ. In [3], we studied the relationship between the weak partition property and the partition property for normal ultrafilters on $P_{κ} λ$ . In this paper we study a hierarchy of properties intermediate between the weak partition property and the partition property. Given appropriate large cardinal assumptions, we show that these properties are not all equivalent.

Weak variants of Martin's Axiom

J. Barnett (1992)

Fundamenta Mathematicae

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Examples exist of smooth maps on the boundary of a smooth manifold M which allow continuous extensions over M without fixed points but no such smooth extensions. Such maps are studied here in more detail. They have a minimal fixed point set when all transversally fixed maps in their homotopy class are considered. Therefore we introduce a Nielsen fixed point theory for transversally fixed maps on smooth manifolds without or with boundary, and use it to calculate the minimum number of...

On a problem of Steve Kalikow

Saharon Shelah (2000)

Fundamenta Mathematicae

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The Kalikow problem for a pair (λ,κ) of cardinal numbers,λ > κ (in particular κ = 2) is whether we can map the family of ω-sequences from λ to the family of ω-sequences from κ in a very continuous manner. Namely, we demand that for η,ν ∈ ω we have: η, ν are almost equal if and only if their images are. We show consistency of the negative answer, e.g., for $ℵ_{ω}$ but we prove it for smaller cardinals. We indicate a close connection with the free subset property and its variants. ...

More topological cardinal inequalities

O. Alas (1993)

Colloquium Mathematicae

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A new topological cardinal invariant is defined; it may be considered as a weaker form of the Lindelöf degree.

Simplicity simplified.

Casanovas, Enrique (2007)

Revista Colombiana de Matemáticas

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Covering of the null ideal may have countable cofinality

Saharon Shelah (2000)

Fundamenta Mathematicae

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We prove that it is consistent that the covering number of the ideal of measure zero sets has countable cofinality.

A generalization of a generic theorem in the theory of cardinal invariants of topological spaces

Alejandro Ramírez-Páramo, Noé Trinidad Tapia-Bonilla (2007)

Commentationes Mathematicae Universitatis Carolinae

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The main goal of this paper is to establish a technical result, which provides an algorithm to prove several cardinal inequalities and relative versions of cardinal inequalities related to the well-known Arhangel’skii’s inequality: If $X$ is a $T_{2}$ -space, then $| X | \leq 2^{L (X) χ (X)}$ . Moreover, we will show relative versions of three well-known cardinal inequalities.