Topology inside ω1


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In this expository paper, we show how the pressing down lemma and Ulam matrices can be used to study the topology of subsets of !1. We prove, for example, that if S and T are stationary subsets of ω1 with SΔT = (S -T)∪ (T - S) stationary, then S and T cannot be homeomorphic. Because Ulam matrices provide !1-many pairwise disjoint stationary subsets of any given stationary set, it follows that there are 2!1-many stationary subsets of any stationary subset of !1 with the property that no two of them are homeomorphic to each other. We also show that if S and T are stationary sets, then the product space S ×T is normal if and only if S ∩ T is stationary. In addition, we prove that for any X ⊆ ω1, X × X is normal, and that if X × X is hereditarily normal, then X × X is metrizable.

Original languageEnglish
Pages (from-to)1989-2000
Number of pages12
JournalRocky Mountain Journal of Mathematics
Issue number6
StatePublished - Dec 2020


  • Borel measure
  • Borel sets Borel measure
  • Club-set
  • Countable ordinals
  • Pressing down lemma
  • Products of stationary sets
  • Stationary set
  • Ulam Matrix
  • ω1


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