Displaying similar documents to “A note on splittable spaces”

Countable compactness and p -limits

Salvador García-Ferreira, Artur Hideyuki Tomita (2001)

Commentationes Mathematicae Universitatis Carolinae

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For M ω * , we say that X is quasi M -compact, if for every f : ω X there is p M such that f ¯ ( p ) X , where f ¯ is the Stone-Čech extension of f . In this context, a space X is countably compact iff X is quasi ω * -compact. If X is quasi M -compact and M is either finite or countable discrete in ω * , then all powers of X are countably compact. Assuming C H , we give an example of a countable subset M ω * and a quasi M -compact space X whose square is not countably compact, and show that in a model of A. Blass and S. Shelah...

Topologies on groups determined by right cancellable ultrafilters

Igor V. Protasov (2009)

Commentationes Mathematicae Universitatis Carolinae

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For every discrete group G , the Stone-Čech compactification β G of G has a natural structure of a compact right topological semigroup. An ultrafilter p G * , where G * = β G G , is called right cancellable if, given any q , r G * , q p = r p implies q = r . For every right cancellable ultrafilter p G * , we denote by G ( p ) the group G endowed with the strongest left invariant topology in which p converges to the identity of G . For any countable group G and any right cancellable ultrafilters p , q G * , we show that G ( p ) is homeomorphic to G ( q ) if...

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 is defined as follows: We say that 𝒜 K if there is a function f : ω ω such that f - 1 ( A ) ( ) for every A ( 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 ( 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 dimensionally restricted maps

H. Murat Tuncali, Vesko Valov (2002)

Fundamenta Mathematicae

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Let f: X → Y be a closed n-dimensional surjective map of metrizable spaces. It is shown that if Y is a C-space, then: (1) the set of all maps g: X → ⁿ with dim(f △ g) = 0 is uniformly dense in C(X,ⁿ); (2) for every 0 ≤ k ≤ n-1 there exists an F σ -subset A k of X such that d i m A k k and the restriction f | ( X A k ) is (n-k-1)-dimensional. These are extensions of theorems by Pasynkov and Toruńczyk, respectively, obtained for finite-dimensional spaces. A generalization of a result due to Dranishnikov and Uspenskij...

General position properties in fiberwise geometric topology

Taras Banakh, Vesko Valov

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General position properties play a crucial role in geometric and infinite-dimensional topologies. Often such properties provide convenient tools for establishing various universality results. One of well-known general position properties is DDⁿ, the property of disjoint n-cells. Each Polish L C n - 1 -space X possessing DDⁿ contains a topological copy of each n-dimensional compact metric space. This fact implies, in particular, the classical Lefschetz-Menger-Nöbeling-Pontryagin-Tolstova embedding...

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 + 1 , n ∈ ℕ, be the n-dimensional Menger continuum in n + 1 , also known as the n-dimensional Sierpiński carpet, and let D be a countable dense subset of M n + 1 . We consider the topological group...

Extension of point-finite partitions of unity

Haruto Ohta, Kaori Yamazaki (2006)

Fundamenta Mathematicae

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A subspace A of a topological space X is said to be P γ -embedded ( P γ (point-finite)-embedded) in X if every (point-finite) partition of unity α on A with |α| ≤ γ extends to a (point-finite) partition of unity on X. The main results are: (Theorem A) A subspace A of X is P γ (point-finite)-embedded in X iff it is P γ -embedded and every countable intersection B of cozero-sets in X with B ∩ A = ∅ can be separated from A by a cozero-set in X. (Theorem B) The product A × [0,1] is P γ (point-finite)-embedded...

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 .