Abstract
An uncountable cardinal κ is called ω1-strongly compact if every κ-complete ultrafilter on any set I can be extended to an ω1-complete ultrafilter on I. We show that the first ω1-strongly compact cardinal, κ0, cannot be a successor cardinal, and that its cofinality is at least the first measurable cardinal. We prove that the Singular Cardinal Hypothesis holds above κ0. We show that the product of Lindel öf spaces is κ-Lindel öf if and only if κ ≥ κ0. Finally, we characterize κ0 in terms of second order reflection for relational structures and we give some applications. For instance, we show that every first-countable nonmetrizable space has a nonmetrizable subspace of size less than κ0.
Original language | English |
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Pages (from-to) | 266-278 |
Number of pages | 13 |
Journal | Journal of Symbolic Logic |
Volume | 79 |
Issue number | 1 |
DOIs | |
State | Published - 2014 |
Bibliographical note
Publisher Copyright:© 2014, Association for Symbolic Logic.
Keywords
- 1-strongly compact cardinal
- Completely regular space
- Countably chromatic graph
- Lindel öf space
- Metrizable space
- Second-order reflection
- Singular Cardinal Hypothesis