A Borel chain condition of T(X)

Abstract

We examine the Borel version of the $$\sigma $$-finite chain condition of Horn and Tarski for a class of posets T(X) which have been used in the solution of their well-known problem. More precisely, we show that the poset $$T(\pi \mathbb{Q} \it )$$ does not have the $$\sigma $$-finite chain condition witnessed by Borel pieces. More precisely, we define a condition on the topological spaces X under which the corresponding Todorcevic ordering T(X) does not have the $$\sigma $$-bounded chain condition witnessed by a countable Borel decomposition although it might satisfy the $$\sigma $$-finite chain condition witnessed by a non Borel decomposition.