An axiomatics of non-Desarguean geometry based on the half-plane as the primitive notion

A. Śniatycki

Displaying similar documents to “An axiomatics of non-Desarguean geometry based on the half-plane as the primitive notion”

The $S_{α}$ separation axioms

J. Guia (1986)

Matematički Vesnik

Similarity:

$T_{i}$ - pairwise continuous functions and bitopological separation axioms

M. Jelić (1989)

Matematički Vesnik

Similarity:

Propositional extensions of $L_{ω} 1_{ω}$

Richard Gostanian, Karel Hrbacek

Similarity:

CONTENTS0. Preliminaries....................................................................... 71. Adding propositional connectives to $L_{ω} 1_{ω}$ ............... 82. The propositional part of $L_{ω} 1_{ω}$ (S)............................. 103. The operation S and the Boolean algebra $B_{S}$ ............... 114. General model-theoretic properties of $L_{ω} 1_{ω}$ (S)...... 175. Hanf number computations...................................................... 226. Negative results for $L_{ω} 1_{ω}$ (S)...........................................

On the differential geometry of some classes of infinite dimensional manifolds

Maysam Maysami Sadr, Danial Bouzarjomehri Amnieh (2024)

Archivum Mathematicum

Similarity:

Albeverio, Kondratiev, and Röckner have introduced a type of differential geometry, which we call lifted geometry, for the configuration space $Γ_{X}$ of any manifold $X$ . The name comes from the fact that various elements of the geometry of $Γ_{X}$ are constructed via lifting of the corresponding elements of the geometry of $X$ . In this note, we construct a general algebraic framework for lifted geometry which can be applied to various “infinite dimensional spaces” associated to $X$ . In order to define...

Axioms weaker then $R_{0}$

J. Guia (1984)

Matematički Vesnik

Similarity:

On the Compactness and Countable Compactness of $2^{ℝ}$ in ZF

Kyriakos Keremedis, Evangelos Felouzis, Eleftherios Tachtsis (2007)

Bulletin of the Polish Academy of Sciences. Mathematics

Similarity:

In the framework of ZF (Zermelo-Fraenkel set theory without the Axiom of Choice) we provide topological and Boolean-algebraic characterizations of the statements " $2^{ℝ}$ is countably compact" and " $2^{ℝ}$ is compact"

How many clouds cover the plane?

James H. Schmerl (2003)

Fundamenta Mathematicae

Similarity:

The plane can be covered by n + 2 clouds iff $2^{ℵ ₀} \leq ℵ ₙ$ .

Essentially Incomparable Banach Spaces of Continuous Functions

Rogério Augusto dos Santos Fajardo (2010)

Bulletin of the Polish Academy of Sciences. Mathematics

Similarity:

We construct, under Axiom ♢, a family ${(C (K_{ξ}))}_{ξ < 2^{(2^{ω})}}$ of indecomposable Banach spaces with few operators such that every operator from $C (K_{ξ})$ into $C (K_{η})$ is weakly compact, for all ξ ≠ η. In particular, these spaces are pairwise essentially incomparable. Assuming no additional set-theoretic axiom, we obtain this result with size $2^{ω}$ instead of $2^{(2^{ω})}$ .

A classification scheme for axioms weaker than $T_{0}$

G. Ervynck (1991)

Matematički Vesnik

Similarity:

On the Set-Theoretic Strength of Countable Compactness of the Tychonoff Product $2^{ℝ}$

Eleftherios Tachtsis (2010)

Bulletin of the Polish Academy of Sciences. Mathematics

Similarity:

We work in ZF set theory (i.e., Zermelo-Fraenkel set theory minus the Axiom of Choice AC) and show the following: 1. The Axiom of Choice for well-ordered families of non-empty sets ( $A C^{W O}$ ) does not imply “the Tychonoff product $2^{ℝ}$ , where 2 is the discrete space 0,1, is countably compact” in ZF. This answers in the negative the following question from Keremedis, Felouzis, and Tachtsis [Bull. Polish Acad. Sci. Math. 55 (2007)]: Does the Countable Axiom of Choice for families of non-empty sets...

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

Horst Herrlich, Eleftherios Tachtsis (2017)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

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...

Coherent ultrafilters and nonhomogeneity

Jan Starý (2015)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

We introduce the notion of a coherent $P$ -ultrafilter on a complete ccc Boolean algebra, strengthening the notion of a $P$ -point on $ω$ , and show that these ultrafilters exist generically under $𝔠 = 𝔡$ . This improves the known existence result of Ketonen [On the existence of $P$ -points in the Stone-Čech compactification of integers, Fund. Math. 92 (1976), 91–94]. Similarly, the existence theorem of Canjar [On the generic existence of special ultrafilters, Proc. Amer. Math. Soc. 110 (1990), no. 1,...

Covering Property Axiom $C P A_{c u b e}$ and its consequences

Krzysztof Ciesielski, Janusz Pawlikowski (2003)

Fundamenta Mathematicae

Similarity:

We formulate a Covering Property Axiom $C P A_{c u b e}$ , which holds in the iterated perfect set model, and show that it implies easily the following facts. (a) For every S ⊂ ℝ of cardinality continuum there exists a uniformly continuous function g: ℝ → ℝ with g[S] = [0,1]. (b) If S ⊂ ℝ is either perfectly meager or universally null then S has cardinality less than . (c) cof() = ω₁ < , i.e., the cofinality of the measure ideal is ω₁. (d) For every uniformly bounded sequence $⟨ f ₙ \in ℝ^{ℝ} ⟩_{n < ω}$ of Borel functions...

On sentences provable in impredicative extensions of theories

Zygmunt Ratajczyk

Similarity:

CONTENTS0. Introduction.......................................................................... 51. Preliminaries............................................................................... 72. Basic facts to be used in the sequel....................................... 113. Predicates OD(.,.) and CL(.,.).................................................... 174. Predicate Sels............................................................................. 185. Strong $\sum_{n}^{1}$ -collection...........................................................

On preimages of ultrafilters in ZF

Horst Herrlich, Paul Howard, Kyriakos Keremedis (2016)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

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...

Operations between sets in geometry

Richard J. Gardner, Daniel Hug, Wolfgang Weil (2013)

Journal of the European Mathematical Society

Similarity:

An investigation is launched into the fundamental characteristics of operations on and between sets, with a focus on compact convex sets and star sets (compact sets star-shaped with respect to the origin) in $n$ -dimensional Euclidean space $ℝ^{n}$ . It is proved that if $n \geq 2$ , with three trivial exceptions, an operation between origin-symmetric compact convex sets is continuous in the Hausdorff metric, $G L (n)$ covariant, and associative if and only if it is $L_{p}$ addition for some $1 \leq p \leq \infty$ . It is also demonstrated...

Cardinal sequences of length < ω₂ under GCH

István Juhász, Lajos Soukup, William Weiss (2006)

Fundamenta Mathematicae

Similarity:

Let (α) denote the class of all cardinal sequences of length α associated with compact scattered spaces (or equivalently, superatomic Boolean algebras). Also put $_{λ} (α) = s \in (α) : s (0) = λ = m i n [s (β) : β < α]$ . We show that f ∈ (α) iff for some natural number n there are infinite cardinals $λ ₀ i > λ ₁ > . . . > λ_{n - 1}$ and ordinals $α ₀, . . ., α_{n - 1}$ such that $α = α ₀ + \dots + α_{n - 1}$ and $f = f ₀ ⏜ f ₁ ⏜ . . . ⏜ f_{n - 1}$ where each $f_{i} \in_{λ_{i}} (α_{i})$ . Under GCH we prove that if α < ω₂ then (i) $_{ω} (α) = s \in^{α} ω, ω ₁ : s (0) = ω$ ; (ii) if λ > cf(λ) = ω, $_{λ} (α) = s \in^{α} λ, λ ⁺ : s (0) = λ, s^{- 1} λ i s ω ₁ - c l o s e d i n α$ ; (iii) if cf(λ) = ω₁, $_{λ} (α) = s \in^{α} λ, λ ⁺ : s (0) = λ, s^{- 1} λ i s ω - c l o s e d a n d s u c c e s s o r - c l o s e d i n α$ ; (iv) if cf(λ) > ω₁, $_{λ} (α) =^{α} λ$ . This yields a complete characterization of the classes (α) for all...

Growth of a primitive of a differential form

Jean-Claude Sikorav (2001)

Bulletin de la Société Mathématique de France

Similarity:

For an exact differential form on a Riemannian manifold to have a primitive bounded by a given function $f$ , by Stokes it has to satisfy some weighted isoperimetric inequality. We show the converse up to some constants if $M$ has bounded geometry. For a volume form, it suffices to have the inequality ( $| Ω | \leq \int_{\partial Ω} f d σ$ for every compact domain $Ω \subset M$ ). This implies in particular the “well-known” result that if $M$ is the universal covering of a compact Riemannian manifold with non-amenable fundamental group, then...

A characterization of sets in $ℝ^{2}$ with DC distance function

Dušan Pokorný, Luděk Zajíček (2022)

Czechoslovak Mathematical Journal

Similarity:

We give a complete characterization of closed sets $F \subset ℝ^{2}$ whose distance function $d_{F} : = dist (\cdot, F)$ is DC (i.e., is the difference of two convex functions on $ℝ^{2}$ ). Using this characterization, a number of properties of such sets is proved.

A tight quantitative version of Arrow’s impossibility theorem

Nathan Keller (2012)

Journal of the European Mathematical Society

Similarity:

The well-known Impossibility Theorem of Arrow asserts that any generalized social welfare function (GSWF) with at least three alternatives, which satisfies Independence of Irrelevant Alternatives (IIA) and Unanimity and is not a dictatorship, is necessarily non-transitive. In 2002, Kalai asked whether one can obtain the following quantitative version of the theorem: For any $ϵ > 0$ , there exists $δ = δ (ϵ)$ such that if a GSWF on three alternatives satisfies the IIA condition and its probability of...