Potential continuity of colorings

Abstract. We say that a coloring c : []n 2 is continuous if it is continuous with respect to some second countable topology on . A coloring c is potentially continuous if it is continuous in some 1-preserving extension of the set-theoretic universe. Given an arbitrary coloring c : []n 2, we define a forcing notion Pc that forces c to be continuous. However, this forcing might collapse cardinals. It turns out that Pc is c.c.c. if and only if Pc is potentially continuous. This gives a combinatorial characterization of potential continuity. On the other hand, we show that adding 1 Cohen reals to any model of set theory introduces a coloring c : [1]2 2 which is potentially continuous but not continuous. 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.

• Type: Journal
• Year: 2008
• Where: AML
• Authors: Stefan Geschke
• Keywords: AML 2008, Arbitrary Coloring, Countable Topology, Set-theoretic Universe