Large strongly anti-Urysohn spaces exist

Juhász, István; Shelah, Saharon; Soukup, Lajos; Szentmiklóssy, Zoltán

doi:10.1016/j.topol.2022.108288

Cited by 1 publication

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“…So, this will show that regularity cannot be weakened to Hausdorff in our main result 3.3. The example is non-trivial, however, fortunately for us, we only need to perform a minor modification of the example from [8] where the hard work was done. Now, the example from [8] is a separable strongly anti-Urysohn (SAU) space X of cardinality 2 c .…”

Section: The Hausdorff Casementioning

confidence: 99%

“…The example is non-trivial, however, fortunately for us, we only need to perform a minor modification of the example from [8] where the hard work was done. Now, the example from [8] is a separable strongly anti-Urysohn (SAU) space X of cardinality 2 c . The SAU property means that any two infinite closed sets in X intersect.…”

Section: The Hausdorff Casementioning

confidence: 99%

“…The reason why the space X from [8] needs to be modified for our purposes is that its weight is 2 c . So, the space we need will be X r , whose topology τ r is the coarser topology on X generated by RO(X), the family of all regular open sets in X.…”

Section: The Hausdorff Casementioning

confidence: 99%

“…The space X from [8] is right-separated, i.e. scattered and, although many consistent examples of crowded SAU spaces had been constructed, their existence in ZFC was not known.…”

Section: The Hausdorff Casementioning

confidence: 99%

“…scattered and, although many consistent examples of crowded SAU spaces had been constructed, their existence in ZFC was not known. So, it was asked explicitly in [8] if they exist. Now, X r has the same isolated points as X, so it is not crowded but using w(X) ≤ c we can actually get such an example, thus giving an affirmative answer to this problem from [8].…”

Section: The Hausdorff Casementioning

confidence: 99%

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Spaces of countable free set number and PFA

Dow

Juhász²

2023

Proc. Amer. Math. Soc.

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The main result of this paper is that, under PFA, for every regular space X X with F ( X ) = ω F(X) = \omega we have | X | ≤ w ( X ) ω |X| \le w(X)^\omega ; in particular, w ( X ) ≤ c w(X) \le \mathfrak {c} implies | X | ≤ c |X| \le \mathfrak {c} . This complements numerous prior results that yield consistent examples of even compact Hausdorff spaces X X with F ( X ) = ω F(X) = \omega such that w ( X ) = c w(X) = \mathfrak {c} and | X | = 2 c |X| = 2^\mathfrak {c} . We also show that regularity cannot be weakened to the Hausdorff property in this result because we can find in ZFC a Hausdorff space X X with F ( X ) = ω F(X) = \omega such that w ( X ) = c w(X) = \mathfrak {c} and | X | = 2 c |X| = 2^\mathfrak {c} . In fact, this space X X has the strongly anti-Urysohn (SAU) property that any two infinite closed sets in X X intersect, which is much stronger than F ( X ) = ω F(X) = \omega . Moreover, any non-empty open set in X X also has size 2 c 2^\mathfrak {c} , and thus our example answers one of the main problems of both Juhász, Soukup, and Szentmiklóssy [Topology Appl. 213 (2016), pp. 8–23] and Juhász, Shelah, Soukup, and Szentmiklóssy [Topology Appl. 323 (2023), Paper No. 108288, 15 pp.] by providing in ZFC a SAU space with no isolated points.

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Section: The Hausdorff Casementioning

confidence: 99%

Section: The Hausdorff Casementioning

confidence: 99%

Section: The Hausdorff Casementioning

confidence: 99%

“…The space X from [8] is right-separated, i.e. scattered and, although many consistent examples of crowded SAU spaces had been constructed, their existence in ZFC was not known.…”

Section: The Hausdorff Casementioning

confidence: 99%

Section: The Hausdorff Casementioning

confidence: 99%

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