A Borel chain condition of T(X)

Abstract

We examine the Borel version of the σ-finite chain condition ofHorn and Tarski for a class of posets T(X) which have been used in the solutionof their well-known problem. More precisely, we show that the poset T(πQ) doesnot have the σ-finite chain condition witnessed by Borel pieces. More precisely,we define a condition on the topological spaces X under which the correspondingTodorcevic ordering T(X) does not have the σ-bounded chain condition witnessedby a countable Borel decomposition although it might satisfy the σ-finite chaincondition witnessed by a non Borel decomposition.