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On Noetherian Spaces22nd Annual IEEE Symposium on Logic in Computer Science (LICS 2007)
- Publisher
- Springer Journals
- Copyright
- Copyright © 2015 by Springer-Verlag Berlin Heidelberg
- Subject
- Mathematics; Mathematical Logic and Foundations; Mathematics, general; Algebra
- ISSN
- 0933-5846
- eISSN
- 1432-0665
- DOI
- 10.1007/s00153-015-0473-4
- Publisher site
- See Article on Publisher Site
Abstract
A quasi-order Q induces two natural quasi-orders on $${\mathcal{P}(Q)}$$ P ( Q ) , but if Q is a well-quasi-order, then these quasi-orders need not necessarily be well-quasi-orders. Nevertheless, Goubault-Larrecq (Proceedings of the 22nd Annual IEEE Symposium 4 on Logic in Computer Science (LICS’07), pp. 453–462, 2007) showed that moving from a well-quasi-order Q to the quasi-orders on $${\mathcal{P}(Q)}$$ P ( Q ) preserves well-quasi-orderedness in a topological sense. Specifically, Goubault-Larrecq proved that the upper topologies of the induced quasi-orders on $${\mathcal{P}(Q)}$$ P ( Q ) are Noetherian, which means that they contain no infinite strictly descending sequences of closed sets. We analyze various theorems of the form “if Q is a well-quasi-order then a certain topology on (a subset of) $${\mathcal{P}(Q)}$$ P ( Q ) is Noetherian” in the style of reverse mathematics, proving that these theorems are equivalent to ACA 0 over RCA 0. To state these theorems in RCA 0 we introduce a new framework for dealing with second-countable topological spaces.
Journal
Archive for Mathematical Logic – Springer Journals
Published: Jan 2, 2016