Ideals without CCC

Journal of Symbolic Logic 63 (1):128-148 (1998)
Let I be an ideal of subsets of a Polish space X, containing all singletons and possessing a Borel basis. Assuming that I does not satisfy ccc, we consider the following conditions (B), (M) and (D). Condition (B) states that there is a disjoint family F $\subseteq$ P(X) of size c, consisting of Borel sets which are not in I. Condition (M) states that there is a Borel function f: X → X with $f^{-1}[\{x\}] \not\in$ I for each x ∈ X. Provided that X is a group and I is invariant, condition (D) states that there exist a Borel set B $\not\in$ I and a perfect set P $\subseteq$ X for which the family $\{B + x: x \in P\}$ is disjoint. The aim of the paper is to study whether the reverse implications in the chain (D) $\Rightarrow$ (M) $\Rightarrow$ (B) $\Rightarrow$ not-ccc can hold. We build a σ-ideal on the Cantor group witnessing (M) & ¬ (D) (Section 2). A modified version of that σ-ideal contains the whole space (Section 3). Some consistency results on deriving (M) from (B) for "nicely" defined ideals are established (Sections 4 and 5). We show that both ccc and (M) can fail (Theorems 1.3 and 5.6). Finally, some sharp versions of (M) for invariant ideals on Polish groups are investigated (Section 6)
Keywords No keywords specified (fix it)
Categories (categorize this paper)
DOI 10.2307/2586592
 Save to my reading list
Follow the author(s)
My bibliography
Export citation
Find it on Scholar
Edit this record
Mark as duplicate
Revision history Request removal from index
Download options
PhilPapers Archive

Upload a copy of this paper     Check publisher's policy on self-archival     Papers currently archived: 24,470
External links
Setup an account with your affiliations in order to access resources via your University's proxy server
Configure custom proxy (use this if your affiliation does not provide a proxy)
Through your library
References found in this work BETA
Jaime I. Ihoda & Saharon Shelah (1989). Δ12-Sets of Reals. Annals of Pure and Applied Logic 42 (3):207-223.

Add more references

Citations of this work BETA

Add more citations

Similar books and articles

Monthly downloads

Added to index


Total downloads

12 ( #355,468 of 1,925,556 )

Recent downloads (6 months)

1 ( #418,152 of 1,925,556 )

How can I increase my downloads?

My notes
Sign in to use this feature

Start a new thread
There  are no threads in this forum
Nothing in this forum yet.