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Chainability And Hemmingsen's Theorem, Paul Bankston
Chainability And Hemmingsen's Theorem, Paul Bankston
Mathematics, Statistics and Computer Science Faculty Research and Publications
On the surface, the definitions of chainability and Lebesgue covering dimension ⩽1 are quite similar as covering properties. Using the ultracoproduct construction for compact Hausdorff spaces, we explore the assertion that the similarity is only skin deep. In the case of dimension, there is a theorem of E. Hemmingsen that gives us a first-order lattice-theoretic characterization. We show that no such characterization is possible for chainability, by proving that if κ is any infinite cardinal and AA is a lattice base for a nondegenerate continuum, then AA is elementarily equivalent to a lattice base for a continuum Y …
The Chang-Los-Suszko Theorem In A Topological Setting, Paul Bankston
The Chang-Los-Suszko Theorem In A Topological Setting, Paul Bankston
Mathematics, Statistics and Computer Science Faculty Research and Publications
The Chang-Łoś-Suszko theorem of first-order model theory characterizes universal-existential classes of models as just those elementary classes that are closed under unions of chains. This theorem can then be used to equate two model-theoretic closure conditions for elementary classes; namely unions of chains and existential substructures. In the present paper we prove a topological analogue and indicate some applications.