On pseudocompactness and related notions in ZF

Kyriakos Keremedis

Displaying similar documents to “On pseudocompactness and related notions in ZF”

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

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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"

Three small results on normal first countable linearly H-closed spaces

Mathieu Baillif (2022)

Commentationes Mathematicae Universitatis Carolinae

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We use topological consequences of , and proved by other authors to show that normal first countable linearly H-closed spaces with various additional properties are compact in these models.

When is $𝐍$ Lindelöf?

Horst Herrlich, George E. Strecker (1997)

Commentationes Mathematicae Universitatis Carolinae

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Theorem. In ZF (i.e., Zermelo-Fraenkel set theory without the axiom of choice) the following conditions are equivalent: (1) $ℕ$ is a Lindelöf space, (2) $ℚ$ is a Lindelöf space, (3) $ℝ$ is a Lindelöf space, (4) every topological space with a countable base is a Lindelöf space, (5) every subspace of $ℝ$ is separable, (6) in $ℝ$ , a point $x$ is in the closure of a set $A$ iff there exists a sequence in $A$ that converges to $x$ , (7) a function $f : ℝ \to ℝ$ is continuous at a point $x$ iff $f$ is sequentially continuous...

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

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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...

A solution to Comfort's question on the countable compactness of powers of a topological group

Artur Hideyuki Tomita (2005)

Fundamenta Mathematicae

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In 1990, Comfort asked Question 477 in the survey book “Open Problems in Topology”: Is there, for every (not necessarily infinite) cardinal number $α \leq 2^{}$ , a topological group G such that $G^{γ}$ is countably compact for all cardinals γ < α, but $G^{α}$ is not countably compact? Hart and van Mill showed in 1991 that α = 2 answers this question affirmatively under $M A_{c o u n t a b l e}$ . Recently, Tomita showed that every finite cardinal answers Comfort’s question in the affirmative, also from $M A_{c o u n t a b l e}$ . However, the question has...

Counting models of set theory

Ali Enayat (2002)

Fundamenta Mathematicae

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Let T denote a completion of ZF. We are interested in the number μ(T) of isomorphism types of countable well-founded models of T. Given any countable order type τ, we are also interested in the number μ(T,τ) of isomorphism types of countable models of T whose ordinals have order type τ. We prove: (1) Suppose ZFC has an uncountable well-founded model and $κ \in ω \cup ℵ ₀, ℵ ₁, 2^{ℵ ₀}$ . There is some completion T of ZF such that μ(T) = κ. (2) If α <ω₁ and μ(T,α) > ℵ₀, then $μ (T, α) = 2^{ℵ ₀}$ . (3) If α < ω₁ and T ⊢ V ≠ OD,...

On topological groups with a small base and metrizability

Saak Gabriyelyan, Jerzy Kąkol, Arkady Leiderman (2015)

Fundamenta Mathematicae

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A (Hausdorff) topological group is said to have a -base if it admits a base of neighbourhoods of the unit, $U_{α} : α \in ℕ^{ℕ}$ , such that $U_{α} \subset U_{β}$ whenever β ≤ α for all $α, β \in ℕ^{ℕ}$ . The class of all metrizable topological groups is a proper subclass of the class $T G_{}$ of all topological groups having a -base. We prove that a topological group is metrizable iff it is Fréchet-Urysohn and has a -base. We also show that any precompact set in a topological group $G \in T G_{}$ is metrizable, and hence G is strictly angelic. We deduce from...

A note on $R_{0}$ -topological spaces

K. K. Dube (1974)

Matematički Vesnik

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Hyperplanes in matroids and the axiom of choice

Marianne Morillon (2022)

Commentationes Mathematicae Universitatis Carolinae

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We show that in set theory without the axiom of choice ZF, the statement sH: “Every proper closed subset of a finitary matroid is the intersection of hyperplanes including it” implies AC $^{fin}$ , the axiom of choice for (nonempty) finite sets. We also provide an equivalent of the statement AC $^{fin}$ in terms of “graphic” matroids. Several open questions stay open in ZF, for example: does sH imply the axiom of choice?

Axioms weaker then $R_{0}$

J. Guia (1984)

Matematički Vesnik

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Remainders of metrizable spaces and a generalization of Lindelöf Σ-spaces

A. V. Arhangel'skii (2011)

Fundamenta Mathematicae

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We establish some new properties of remainders of metrizable spaces. In particular, we show that if the weight of a metrizable space X does not exceed $2^{ω}$ , then any remainder of X in a Hausdorff compactification is a Lindelöf Σ-space. An example of a metrizable space whose remainder in some compactification is not a Lindelöf Σ-space is given. A new class of topological spaces naturally extending the class of Lindelöf Σ-spaces is introduced and studied. This leads to the following theorem:...

The gap between I₃ and the wholeness axiom

Paul Corazza (2003)

Fundamenta Mathematicae

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∃κI₃(κ) is the assertion that there is an elementary embedding $i : V_{λ} \to V_{λ}$ with critical point below λ, and with λ a limit. The Wholeness Axiom, or WA, asserts that there is a nontrivial elementary embedding j: V → V; WA is formulated in the language ∈,j and has as axioms an Elementarity schema, which asserts that j is elementary; a Critical Point axiom, which asserts that there is a least ordinal moved by j; and includes every instance of the Separation schema for j-formulas. Because no instance...

Nonnormality of remainders of some topological groups

Aleksander V. Arhangel&#039;skii, J. van Mill (2016)

Commentationes Mathematicae Universitatis Carolinae

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It is known that every remainder of a topological group is Lindelöf or pseudocompact. Motivated by this result, we study in this paper when a topological group $G$ has a normal remainder. In a previous paper we showed that under mild conditions on $G$ , the Continuum Hypothesis implies that if the Čech-Stone remainder $G^{*}$ of $G$ is normal, then it is Lindelöf. Here we continue this line of investigation, mainly for the case of precompact groups. We show that no pseudocompact group, whose weight...

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

M. Jelić (1989)

Matematički Vesnik

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On star covering properties related to countable compactness and pseudocompactness

Marcelo D. Passos, Heides L. Santana, Samuel G. da Silva (2017)

Commentationes Mathematicae Universitatis Carolinae

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We prove a number of results on star covering properties which may be regarded as either generalizations or specializations of topological properties related to the ones mentioned in the title of the paper. For instance, we give a new, entirely combinatorial proof of the fact that $Ψ$ -spaces constructed from infinite almost disjoint families are not star-compact. By going a little further we conclude that if $X$ is a star-compact space within a certain class, then $X$ is neither first countable...

Lindelöf indestructibility, topological games and selection principles

Marion Scheepers, Franklin D. Tall (2010)

Fundamenta Mathematicae

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Arhangel’skii proved that if a first countable Hausdorff space is Lindelöf, then its cardinality is at most $2^{ℵ ₀}$ . Such a clean upper bound for Lindelöf spaces in the larger class of spaces whose points are $G_{δ}$ has been more elusive. In this paper we continue the agenda started by the second author, [Topology Appl. 63 (1995)], of considering the cardinality problem for spaces satisfying stronger versions of the Lindelöf property. Infinite games and selection principles, especially the Rothberger...

Ordinals in topological groups

Raushan Z. Buzyakova (2007)

Fundamenta Mathematicae

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We show that if an uncountable regular cardinal τ and τ + 1 embed in a topological group G as closed subspaces then G is not normal. We also prove that an uncountable regular cardinal cannot be embedded in a torsion free Abelian group that is hereditarily normal. These results are corollaries to our main results about ordinals in topological groups. To state the main results, let τ be an uncountable regular cardinal and G a T₁ topological group. We prove, among others, the following...

Linear extenders and the Axiom of Choice

Marianne Morillon (2017)

Commentationes Mathematicae Universitatis Carolinae

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In set theory without the Axiom of Choice ZF, we prove that for every commutative field $𝕂$ , the following statement $𝐃_{𝕂}$ : “On every non null $𝕂$ -vector space, there exists a non null linear form” implies the existence of a “ $𝕂$ -linear extender” on every vector subspace of a $𝕂$ -vector space. This solves a question raised in Morillon M., Linear forms and axioms of choice, Comment. Math. Univ. Carolin. 50 (2009), no. 3, 421-431. In the second part of the paper, we generalize our results in the case...