Topologies on groups determined by right cancellable ultrafilters

Igor V. Protasov

Displaying similar documents to “Topologies on groups determined by right cancellable ultrafilters”

A compact Hausdorff topology that is a T₁-complement of itself

Dmitri Shakhmatov, Michael Tkachenko (2002)

Fundamenta Mathematicae

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Topologies τ₁ and τ₂ on a set X are called T₁-complementary if τ₁ ∩ τ₂ = X∖F: F ⊆ X is finite ∪ ∅ and τ₁∪τ₂ is a subbase for the discrete topology on X. Topological spaces $(X, τ_{X})$ and $(Y, τ_{Y})$ are called T₁-complementary provided that there exists a bijection f: X → Y such that $τ_{X}$ and $f^{- 1} (U) : U \in τ_{Y}$ are T₁-complementary topologies on X. We provide an example of a compact Hausdorff space of size $2^{}$ which is T₁-complementary to itself ( denotes the cardinality of the continuum). We prove that the existence of a compact...

MAD families and $P$ -points

Salvador García-Ferreira, Paul J. Szeptycki (2007)

Commentationes Mathematicae Universitatis Carolinae

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The Katětov ordering of two maximal almost disjoint (MAD) families $𝒜$ and $ℬ$ is defined as follows: We say that $𝒜 \leq_{K} ℬ$ if there is a function $f : ω \to ω$ such that $f^{- 1} (A) \in ℐ (ℬ)$ for every $A \in ℐ (𝒜)$ . In [Garcia-Ferreira S., Hrušák M., Ordering MAD families a la Katětov, J. Symbolic Logic 68 (2003), 1337–1353] a MAD family is called $K$ -uniform if for every $X \in ℐ {(𝒜)}^{+}$ , we have that ${𝒜 |}_{X} \leq_{K} 𝒜$ . We prove that CH implies that for every $K$ -uniform MAD family $𝒜$ there is a $P$ -point $p$ of $ω^{*}$ such that the set of all Rudin-Keisler predecessors of $p$ is dense...

$ω$ H-sets and cardinal invariants

Alessandro Fedeli (1998)

Commentationes Mathematicae Universitatis Carolinae

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A subset $A$ of a Hausdorff space $X$ is called an $ω$ H-set in $X$ if for every open family $𝒰$ in $X$ such that $A \subset ⋃ 𝒰$ there exists a countable subfamily $𝒱$ of $𝒰$ such that $A \subset ⋃ {\bar{V} : V \in 𝒱}$ . In this paper we introduce a new cardinal function $t_{s θ}$ and show that $| A | \leq 2^{t_{s θ} (X) ψ_{c} (X)}$ for every $ω$ H-set $A$ of a Hausdorff space $X$ .

More reflections on compactness

Lúcia R. Junqueira, Franklin D. Tall (2003)

Fundamenta Mathematicae

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We consider the question of when $X_{M} = X$ , where $X_{M}$ is the elementary submodel topology on X ∩ M, especially in the case when $X_{M}$ is compact.

A countably cellular topological group all of whose countable subsets are closed need not be $ℝ$ -factorizable

Mihail G. Tkachenko (2023)

Commentationes Mathematicae Universitatis Carolinae

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We construct a Hausdorff topological group $G$ such that $ℵ_{1}$ is a precalibre of $G$ (hence, $G$ has countable cellularity), all countable subsets of $G$ are closed and $C$ -embedded in $G$ , but $G$ is not $ℝ$ -factorizable. This solves Problem 8.6.3 from the book “Topological Groups and Related Structures" (2008) in the negative.

A MAD Q-set

Arnold W. Miller (2003)

Fundamenta Mathematicae

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A MAD (maximal almost disjoint) family is an infinite subset of the infinite subsets of ω = 0,1,2,... such that any two elements of intersect in a finite set and every infinite subset of ω meets some element of in an infinite set. A Q-set is an uncountable set of reals such that every subset is a relative $G_{δ}$ -set. It is shown that it is relatively consistent with ZFC that there exists a MAD family which is also a Q-set in the topology it inherits as a subset of $P (ω) = 2^{ω}$ .

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...

Homeomorphism groups of Sierpiński carpets and Erdős space

Jan J. Dijkstra, Dave Visser (2010)

Fundamenta Mathematicae

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Erdős space is the “rational” Hilbert space, that is, the set of vectors in ℓ² with all coordinates rational. Erdős proved that is one-dimensional and homeomorphic to its own square × , which makes it an important example in dimension theory. Dijkstra and van Mill found topological characterizations of . Let $M ₙ^{n + 1}$ , n ∈ ℕ, be the n-dimensional Menger continuum in $ℝ^{n + 1}$ , also known as the n-dimensional Sierpiński carpet, and let D be a countable dense subset of $M ₙ^{n + 1}$ . We consider the topological group...