Totally non-remote points in $\beta \mathbb {Q}$

Sergei Logunov

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Compacta are maximally $G_{δ}$ -resolvable

István Juhász, Zoltán Szentmiklóssy (2013)

Commentationes Mathematicae Universitatis Carolinae

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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

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On the set $ℝ$ of real numbers we consider a poset $𝒫_{τ} (ℝ)$ (by inclusion) of topologies $τ (A)$ , where $A \subseteq ℝ$ , such that $A_{1} \supseteq A_{2}$ iff $τ (A_{1}) \subseteq τ (A_{2})$ . The poset has the minimal element $τ (ℝ)$ , the Euclidean topology, and the maximal element $τ (\emptyset)$ , the Sorgenfrey topology. We are interested when two topologies $τ_{1}$ and $τ_{2}$ (especially, for $τ_{2} = τ (\emptyset)$ ) 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 $(ℝ, τ (\emptyset))$ iff $A$ is countable. We study also common...

An independency result in connectification theory

Alessandro Fedeli, Attilio Le Donne (1999)

Commentationes Mathematicae Universitatis Carolinae

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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 $\neg ψ$ is provable in ZFC.

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

Compacta are maximally G δ -resolvable

A poset of topologies on the set of real numbers

An independency result in connectification theory

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

Compacta are maximally $G_{δ}$ -resolvable