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More about spaces with a small diagonal

Alan Dow, Oleg Pavlov

2006
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Fundamenta Mathematicae
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Hušek defines a space X to have a small diagonal if each uncountable subset of X 2 disjoint from the diagonal, has an uncountable subset whose closure is disjoint from the diagonal. Hušek proved that a compact space of weight ω 1 which has a small diagonal, will be metrizable, but it remains an open problem to determine if the weight restriction is necessary. It has been shown to be consistent that each compact space with a small diagonal is metrizable, in particular, Juhasz and Szentmiklossy
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... and Szentmiklossy [JS92] proved that this holds in models of CH. In this paper we prove that it also follows from the Proper Forcing Axiom (PFA). We also present two (consistent) examples of countably compact non-metrizable spaces with small diagonal, one of which maps perfectly onto ω 1 . Proposition 2. [JS92] A compact space with a small diagonal will have countable tightness. Proposition 3. [Dow88b] A countably compact space is metrizable if each of its subspaces of cardinality at most ℵ 1 is metrizable. Corollary 5. If a compact space has a small diagonal, then it is metrizable if each of its separable subspaces is metrizable. Proof. By Proposition 3, we may assume that X has a dense subset of cardinality at most ℵ 1 and, by 2, that X has countable tightness. Therefore X can be written as an increasing union of compact separable subspaces. If each of these is metrizable, then each has countable weight. In addition, X would then have a net of cardinality ℵ 1 . Since the weight of a compact space is equal to the minimum cardinaity of a net (i.e. weight is equal to net weight), we would have that X has weight at most ω 1 , and so by Proposition 4, X is metrizable. Proposition 6. [Gru02] A first-countable hereditarily Lindelöf space X with a small diagonal, will have a G δ -diagonal. PFA and compact spaces with small diagonal As is well-known, it follows from PFA that there is no S-space. Proposition 7. [Tod89] PFA implies that each hereditarily separable (hS) space is also hereditarily Lindelöf (hL). Proposition 8. [Tod89] PFA implies OCA: if X is a separable metric space and K 0 is a symmetric open subset of X 2 \ ∆ X , then either there is an uncountable Y ⊂ X such that Y 2 \∆ X ⊂ K 0 , or, X can be covered by a countable family {X n : n ∈ ω} such that for each n, X 2 n is disjoint from K 0 . Our main result of this section is the following theorem. Theorem 9. PFA implies that each compact space with a small diagonal is metrizable. By Corollary 5, we may assume that our space of interest is separable and we will use the following characterization from [Gru02, 1.2].

doi:10.4064/fm191-1-5
fatcat:o33krlkopncgfe6xk2os7b5x4i