Anti-Urysohn spaces

Abstract

Abstract. All spaces are assumed to be infinite Hausdorff spaces. We call a space anti-Urysohn (AU in short) iff any two non-emty regular closed sets in it intersect. We prove that• for every infinite cardinal κ there is a space of size κ in which fewer than cf (κ) many non-empty regular closed sets always intersect; • there is a locally countable AU space of size κ iff ω ≤ κ ≤ 2 c . A space with at least two non-isolated points is called strongly anti-Urysohn (SAU in short) iff any two infinite closed sets in it intersect. We prove that• if X is any SAU space then s ≤ |X| ≤ 2 2 c ;

Large strongly anti-Urysohn spaces exist

“…However, it remains an open question if 2 c is an upper bound for the sizes of all SAU spaces. It was proved in [2] that 2 2 c is such an upper bound.…”

“…Anti-Urysohn (AU) and strongly anti-Urysohn (SAU) spaces were introduced and studied in [2]. An AU space is a Hausdorff space in which any two non-empty regular closed sets intersect and a SAU space is a Hausdorff space that has at least two non-isolated points and in which any two infinite closed sets intersect.…”

“…All relevant questions concerning AU spaces were settled in [2], in particular it was shown that for every infinite cardinal κ there is an AU space of cardinality κ, but only inconclusive partial results were proved for SAU spaces. For instance, we could only construct consistent examples of SAU spaces, moreover all of them were of size ≤ c, while only 2 2 c was established as an upper bound for their cardinality.…”

“…It was proved in [2,Theorem 3.7] that r = c implies the existence of a locally countable and crowded SAU space of size ≤ c, moreover several other forcing constructions yielded crowded SAU spaces. This then led to the question if the existence of a SAU space is equivalent to the existence of crowded ones.…”

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As defined in [2], a Hausdorff space is strongly anti-Urysohn (in short: SAU) if it has at least two non-isolated points and any two infinite closed subsets of it intersect. Our main result answers the two main questions of [2] by providing a ZFC construction of a locally countable SAU space of cardinality 2 c .

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