Compact Hausdorff spaces with two open sets

Mioduszewski, J.

doi:10.4064/cm-39-1-35-40

Cited by 2 publications

(2 citation statements)

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“…Spaces which only have λ different homeomorphism types amongst their open subspaces (for some cardinal λ) are said to be of diversity λ [19]. One checks that Lemma 4.7 applies to all compact Hausdorff spaces of diversity two, which are known to be zero-dimensional [17]. In particular, Lemma 4.7 applies to the Cantor space C, which can be characterised as the unique compact metrizable space of diversity two [26].…”

Section: Finite Compactifications Of Cards Of ω * and S κmentioning

confidence: 99%

The space $ω^*$ and the reconstruction of normality

Pitz

2015

Preprint

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The topological reconstruction problem asks how much information about a topological space can be recovered from its point-complement subspaces. If the whole space can be recovered in this way, it is called reconstructible.Our main result states that it is independent of the axioms of set theory (ZFC) whether the Stone-Čech remainder of the integers ω * is reconstructible. Our second result is about the reconstruction of normality. We show that assuming the Continuum Hypothesis, the compact Hausdorff space ω * has a non-normal reconstruction, namely the space ω * \ {p} for a P -point p of ω * . More generally, we show that the existence of an uncountable cardinal κ satisfying κ = κ <κ implies that there is a normal space with a non-normal reconstruction.These results demonstrate that consistently, the property of being a normal space is not reconstructible. Whether normality is non-reconstructible in ZFC remains an open question.

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Section: Finite Compactifications Of Cards Of ω * and S κmentioning

confidence: 99%

The space $ω^*$ and the reconstruction of normality

Pitz

2015

Preprint

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“…Spaces which only have λ different homeomorphism types amongst their open subspaces (for some cardinal λ) are said to be of diversity λ [8]. One checks that Lemma 2.1 applies to all compact Hausdorff spaces of diversity two, which are known to be zero-dimensional [7]. In particular, it applies to the Cantor space C, which can be characterised as the unique compact metrizable space of diversity two [10], and to the Alexandroff Double Arrow space D and also to their product D ×C.…”

Section: Theory and First Examplesmentioning

confidence: 99%

Finite compactifications of $ω^* \setminus \{x\}$

Pitz¹,

Suabedissen²

2013

Preprint

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We prove that under [CH], finite compactifications of ω * \ {x} are homeomorphic to ω * . Moreover, in each case, the remainder consists almost exclusively of P -points, apart from possibly one point.Similar results are obtained for other, related classes of spaces, amongst them Sκ, the κ-Parovičenko space of weight κ. Also, some parallels are drawn to the Cantor set and the Double Arrow space.

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Compact Hausdorff spaces with two open sets

Cited by 2 publications

References 0 publications

The space $ω^*$ and the reconstruction of normality

The space $ω^*$ and the reconstruction of normality

Finite compactifications of $ω^* \setminus \{x\}$

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