December 2020 Topology inside $\omega_1$
Harold Bennett, David Lutzer
Rocky Mountain J. Math. 50(6): 1989-2000 (December 2020). DOI: 10.1216/rmj.2020.50.1989


In this expository paper, we show how the pressing down lemma and Ulam matrices can be used to study the topology of subsets of ω1. We prove, for example, that if S and T are stationary subsets of ω1 with SΔT=(ST)(TS) stationary, then S and T cannot be homeomorphic. Because Ulam matrices provide ω1-many pairwise disjoint stationary subsets of any given stationary set, it follows that there are 2ω1-many stationary subsets of any stationary subset of ω1 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 ST is stationary. In addition, we prove that for any Xω1, X×X is normal, and that if X×X is hereditarily normal, then X×X is metrizable.


Download Citation

Harold Bennett. David Lutzer. "Topology inside $\omega_1$." Rocky Mountain J. Math. 50 (6) 1989 - 2000, December 2020.


Received: 14 January 2020; Accepted: 26 May 2020; Published: December 2020
First available in Project Euclid: 5 January 2021

Digital Object Identifier: 10.1216/rmj.2020.50.1989

Primary: 54B10 , 54F05 , 54G15
Secondary: 03E10

Keywords: $\omega_1$ , Borel measure , Borel sets Borel measure , club-set , countable ordinals , pressing down lemma , products of stationary sets , stationary set , Ulam matrix

Rights: Copyright © 2020 Rocky Mountain Mathematics Consortium


This article is only available to subscribers.
It is not available for individual sale.

Vol.50 • No. 6 • December 2020
Back to Top