Displaying similar documents to “Totally non-remote points in β ℚ ”

Compacta are maximally G δ -resolvable

István Juhász, Zoltán Szentmiklóssy (2013)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

It is well-known that compacta (i.e. compact Hausdorff spaces) are maximally resolvable, that is every compactum X contains Δ ( X ) many pairwise disjoint dense subsets, where Δ ( X ) denotes the minimum size of a non-empty open set in X . The aim of this note is to prove the following analogous result: Every compactum X contains Δ δ ( X ) many pairwise disjoint G δ -dense subsets, where Δ δ ( X ) denotes the minimum size of a non-empty G δ set in X .

A poset of topologies on the set of real numbers

Vitalij A. Chatyrko, Yasunao Hattori (2013)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

On the set ℝ of real numbers we consider a poset 𝒫 τ ( ℝ ) (by inclusion) of topologies τ ( A ) , where A ⊆ ℝ , such that A 1 ⊇ A 2 iff τ ( A 1 ) ⊆ τ ( A 2 ) . The poset has the minimal element τ ( ℝ ) , the Euclidean topology, and the maximal element τ ( ∅ ) , the Sorgenfrey topology. We are interested when two topologies τ 1 and τ 2 (especially, for τ 2 = τ ( ∅ ) ) from the poset define homeomorphic spaces ( ℝ , τ 1 ) and ( ℝ , τ 2 ) . In particular, we prove that for a closed subset A of ℝ the space ( ℝ , τ ( A ) ) is homeomorphic to the Sorgenfrey line ( ℝ , τ ( ∅ ) ) iff A is countable. We study also common...

An independency result in connectification theory

Alessandro Fedeli, Attilio Le Donne (1999)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

A space is called connectifiable if it can be densely embedded in a connected Hausdorff space. Let ψ be the following statement: “a perfect T 3 -space X with no more than 2 𝔠 clopen subsets is connectifiable if and only if no proper nonempty clopen subset of X is feebly compact". In this note we show that neither ψ nor ¬ ψ is provable in ZFC.

Topologies on groups determined by right cancellable ultrafilters

Igor V. Protasov (2009)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

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...