Potential continuity of colorings

Archive for Mathematical Logic 47 (6):567-578 (2008)
  Copy   BIBTEX


We say that a coloring ${c: [\kappa]^n\to 2}$ is continuous if it is continuous with respect to some second countable topology on κ. A coloring c is potentially continuous if it is continuous in some ${\aleph_1}$ -preserving extension of the set-theoretic universe. Given an arbitrary coloring ${c:[\kappa]^n\to 2}$ , we define a forcing notion ${\mathbb P_c}$ that forces c to be continuous. However, this forcing might collapse cardinals. It turns out that ${\mathbb P_c}$ is c.c.c. if and only if c is potentially continuous. This gives a combinatorial characterization of potential continuity. On the other hand, we show that adding ${\aleph_1}$ Cohen reals to any model of set theory introduces a coloring ${c:[\aleph_1]^2 \to 2}$ which is potentially continuous but not continuous. ${\aleph_1}$ has no uncountable c-homogeneous subset in the Cohen extension, but such a set can be introduced by forcing. The potential continuity of c can be destroyed by some c.c.c. forcing



    Upload a copy of this work     Papers currently archived: 92,953

External links

Setup an account with your affiliations in order to access resources via your University's proxy server

Through your library

Similar books and articles

σ-Continuity and related forcings.Marcin Sabok - 2009 - Archive for Mathematical Logic 48 (5):449-464.
Degrees of Unsolvability of Continuous Functions.Joseph S. Miller - 2004 - Journal of Symbolic Logic 69 (2):555 - 584.
Forcing notions in inner models.David Asperó - 2009 - Archive for Mathematical Logic 48 (7):643-651.
A Very Discontinuous Borel Function.Juris Steprans - 1994 - Journal of Symbolic Logic 59 (4):1268-1283.
A model with no magic set.Krzysztof Ciesielski & Saharon Shelah - 1999 - Journal of Symbolic Logic 64 (4):1467-1490.
Knaster and Friends III: Subadditive Colorings.Chris Lambie-Hanson & Assaf Rinot - 2023 - Journal of Symbolic Logic 88 (3):1230-1280.
Partitions of large Rado graphs.M. Džamonja, J. A. Larson & W. J. Mitchell - 2009 - Archive for Mathematical Logic 48 (6):579-606.


Added to PP

29 (#568,210)

6 months
3 (#1,045,430)

Historical graph of downloads
How can I increase my downloads?