OPEN COVERS AND THE SQUARE BRACKET PARTITION RELATION

An open cover U of an infinite separable metric space X is an omega-co ver of X if X is not an element of U and for every finite subset F of X there is a U is an element of U such that F subset of or equal to U. Let Omega be the collection of omega-covers of X. We show that the pa rtition relation Omega --> [Omega](2)(2) holds if, (and only if, the p artition relation Omega --> [Omega](3)(2) holds.