Covering Property Axiom $CPA_{cube}$ and its consequences

Krzysztof Ciesielski; Janusz Pawlikowski

Displaying similar documents to “Covering Property Axiom $C P A_{c u b e}$ and its consequences”

On the solvability of systems of linear equations over the ring $ℤ$ of integers

Horst Herrlich, Eleftherios Tachtsis (2017)

Commentationes Mathematicae Universitatis Carolinae

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We investigate the question whether a system ${(E_{i})}_{i \in I}$ of homogeneous linear equations over $ℤ$ is non-trivially solvable in $ℤ$ provided that each subsystem ${(E_{j})}_{j \in J}$ with $| J | \leq c$ is non-trivially solvable in $ℤ$ where $c$ is a fixed cardinal number such that $c < | I |$ . Among other results, we establish the following. (a) The answer is ‘No’ in the finite case (i.e., $I$ being finite). (b) The answer is ‘No’ in the denumerable case (i.e., $| I | = ℵ_{0}$ and $c$ a natural number). (c) The answer in case that $I$ is uncountable and $c \leq ℵ_{0}$ is ‘No...

On certain non-constructive properties of infinite-dimensional vector spaces

Eleftherios Tachtsis (2018)

Commentationes Mathematicae Universitatis Carolinae

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In set theory without the axiom of choice ( $AC$ ), we study certain non-constructive properties of infinite-dimensional vector spaces. Among several results, we establish the following: (i) None of the principles AC $^{LO}$ (AC for linearly ordered families of nonempty sets)—and hence AC $^{WO}$ (AC for well-ordered families of nonempty sets)— $DC (< κ)$ (where $κ$ is an uncountable regular cardinal), and “for every infinite set $X$ , there is a bijection $f : X \to {0, 1} \times X$ ”, implies the statement “there exists a field $F$ such that...

On preimages of ultrafilters in ZF

Horst Herrlich, Paul Howard, Kyriakos Keremedis (2016)

Commentationes Mathematicae Universitatis Carolinae

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We show that given infinite sets $X, Y$ and a function $f : X \to Y$ which is onto and $n$ -to-one for some $n \in ℕ$ , the preimage of any ultrafilter $ℱ$ of $Y$ under $f$ extends to an ultrafilter. We prove that the latter result is, in some sense, the best possible by constructing a permutation model $ℳ$ with a set of atoms $A$ and a finite-to-one onto function $f : A \to ω$ such that for each free ultrafilter of $ω$ its preimage under $f$ does not extend to an ultrafilter. In addition, we show that in $ℳ$ there exists an ultrafilter compact...

Some generalizations of Olivier's theorem

Alain Faisant, Georges Grekos, Ladislav Mišík (2016)

Mathematica Bohemica

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Let $\sum_{n = 1}^{\infty} a_{n}$ be a convergent series of positive real numbers. L. Olivier proved that if the sequence $(a_{n})$ is non-increasing, then $lim_{n \to \infty} n a_{n} = 0$ . In the present paper: (a) We formulate and prove a necessary and sufficient condition for having $lim_{n \to \infty} n a_{n} = 0$ ; Olivier’s theorem is a consequence of our Theorem . (b) We prove properties analogous to Olivier’s property when the usual convergence is replaced by the $ℐ$ -convergence, that is a convergence according to an ideal $ℐ$ of subsets of $ℕ$ . Again, Olivier’s theorem is a consequence...

Spaces with property $(D C (ω_{1}))$

Wei-Feng Xuan, Wei-Xue Shi (2017)

Commentationes Mathematicae Universitatis Carolinae

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We prove that if $X$ is a first countable space with property $(D C (ω_{1}))$ and with a $G_{δ}$ -diagonal then the cardinality of $X$ is at most $𝔠$ . We also show that if $X$ is a first countable, DCCC, normal space then the extent of $X$ is at most $𝔠$ .

A note on the multiplier ideals of monomial ideals

Cheng Gong, Zhongming Tang (2015)

Czechoslovak Mathematical Journal

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Let $𝔞 \subseteq ℂ [x_{1}, ..., x_{n}]$ be a monomial ideal and $𝒥 (𝔞^{c})$ the multiplier ideal of $𝔞$ with coefficient $c$ . Then $𝒥 (𝔞^{c})$ is also a monomial ideal of $ℂ [x_{1}, ..., x_{n}]$ , and the equality $𝒥 (𝔞^{c}) = 𝔞$ implies that $0 < c < n + 1$ . We mainly discuss the problem when $𝒥 (𝔞) = 𝔞$ or $𝒥 (𝔞^{n + 1 - ε}) = 𝔞$ for all $0 < ε < 1$ . It is proved that if $𝒥 (𝔞) = 𝔞$ then $𝔞$ is principal, and if $𝒥 (𝔞^{n + 1 - ε}) = 𝔞$ holds for all $0 < ε < 1$ then $𝔞 = (x_{1}, ..., x_{n})$ . One global result is also obtained. Let $\tilde{𝔞}$ be the ideal sheaf on $ℙ^{n - 1}$ associated with $𝔞$ . Then it is proved that the equality $𝒥 (\tilde{𝔞}) = \tilde{𝔞}$ implies that $\tilde{𝔞}$ is principal.

Balcar's theorem on supports

Lev Bukovský (2018)

Commentationes Mathematicae Universitatis Carolinae

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In A theorem on supports in the theory of semisets [Comment. Math. Univ. Carolinae 14 (1973), no. 1, 1–6] B. Balcar showed that if $σ \subseteq D \in M$ is a support, $M$ being an inner model of ZFC, and $𝒫 (D ∖ σ) \cap M = r ` ` σ$ with $r \in M$ , then $r$ determines a preorder " $⪯$ " of $D$ such that $σ$ becomes a filter on $(D, ⪯)$ generic over $M$ . We show that if the relation $r$ is replaced by a function $𝒫 (D ∖ σ) \cap M = f_{- 1} (σ)$ , then there exists an equivalence relation " $\sim$ " on $D$ and a partial order on $D / \sim$ such that $D / \sim$ is a complete Boolean algebra, $σ / \sim$ is a generic filter and ${[f (u)]}_{\sim} = - \sum (u / \sim)$ for...

Complete pairs of coanalytic sets

Jean Saint Raymond (2007)

Fundamenta Mathematicae

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Let X be a Polish space, and let C₀ and C₁ be disjoint coanalytic subsets of X. The pair (C₀,C₁) is said to be complete if for every pair (D₀,D₁) of disjoint coanalytic subsets of $ω^{ω}$ there exists a continuous function $f : ω^{ω} \to X$ such that $f^{- 1} (C ₀) = D ₀$ and $f^{- 1} (C ₁) = D ₁$ . We give several explicit examples of complete pairs of coanalytic sets.

On hereditary normality of $ω^{*}$ , Kunen points and character $ω_{1}$

Sergei Logunov (2021)

Commentationes Mathematicae Universitatis Carolinae

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We show that $ω^{*} ∖ {p}$ is not normal, if $p$ is a limit point of some countable subset of $ω^{*}$ , consisting of points of character $ω_{1}$ . Moreover, such a point $p$ is a Kunen point and a super Kunen point.

Displaying similar documents to “Covering Property Axiom C P A c u b e and its consequences”

Displaying similar documents to “Covering Property Axiom $C P A_{c u b e}$ and its consequences”