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PFA implies that there are no S-spaces
Statement
In ZFC plus PFA, no regular Hausdorff hereditarily separable non-Lindelöf space exists; equivalently, there are no S-spaces.
Facts & Assumptions
Given: PFA.
PFA implies PID. PFA implies the P-ideal dichotomy
PID together with makes every regular Hausdorff hereditarily separable space hereditarily Lindelöf, excluding S-spaces. PID plus p greater than omega-one eliminates S-spaces
Proof
By F1 and F2, the PFA universe satisfies both PID and .
Apply F3. Every regular Hausdorff hereditarily separable space is hereditarily Lindelöf and therefore Lindelöf itself, so none meets the non-Lindelöf clause in the definition of an S-space.
Depends on
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
9 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- Todorcevic, Forcing with a coherent Souslin tree, Section 7 (standard reference, not scraped)
- Todorcevic, Combinatorial Dichotomies in Set Theory, Theorem 23.2 (standard reference, not scraped)