Displaying similar documents to “Filling boxes densely and disjointly”

The Golomb space is topologically rigid

Taras O. Banakh, Dario Spirito, Sławomir Turek (2021)

Commentationes Mathematicae Universitatis Carolinae

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The Golomb space ℕ τ is the set ℕ of positive integers endowed with the topology τ generated by the base consisting of arithmetic progressions { a + b n : n ≥ 0 } with coprime a , b . We prove that the Golomb space ℕ τ is topologically rigid in the sense that its homeomorphism group is trivial. This resolves a problem posed by T. Banakh at Mathoverflow in 2017.

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 ≤ K ℬ if there is a function f : ω → ω such that f - 1 ( A ) ∈ ℐ ( ℬ ) for every A ∈ 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 ∈ ℐ ( 𝒜 ) + , we have that 𝒜 | X ≤ 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...

On continuous self-maps and homeomorphisms of the Golomb space

Taras O. Banakh, Jerzy Mioduszewski, Sławomir Turek (2018)

Commentationes Mathematicae Universitatis Carolinae

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The Golomb space ℕ τ is the set ℕ of positive integers endowed with the topology τ generated by the base consisting of arithmetic progressions { a + b n : n ≥ 0 } with coprime a , b . We prove that the Golomb space ℕ τ has continuum many continuous self-maps, contains a countable disjoint family of infinite closed connected subsets, the set Π of prime numbers is a dense metrizable subspace of ℕ τ , and each homeomorphism h of ℕ τ has the following properties: h ( 1 ) = 1 , h ( Π ) = Π , Π h ( x ) = h ( Π x ) , and h ( x ℕ ) = h ( x ) ℕ for all x ∈ ℕ . Here x ℕ : = { x n : n ∈ ℕ } and Π x denotes the set of...

An irrational problem

Franklin D. Tall (2002)

Fundamenta Mathematicae

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Given a topological space ⟨X,⟩ ∈ M, an elementary submodel of set theory, we define X M to be X ∩ M with topology generated by U ∩ M : U ∈ ∩ M . Suppose X M is homeomorphic to the irrationals; must X = X M ? We have partial results. We also answer a question of Gruenhage by showing that if X M is homeomorphic to the “Long Cantor Set”, then X = X M .