An answer to Furstenbergs problem on topological disjointness

In this paper we give an answer to Furstenbergs problem on topological disjointness. Namely, we show that a transitive system $(X,T)$ is disjoint from all minimal systems if and only if $(X,T)$ is weakly mixing and there is some countable dense subset $D$ of $X$ such that for any minimal system $(Y,S)$, any point $yin Y$ and any open neighbourhood $V$ of $y$, and for any nonempty open subset $Usubset X$, there is $xin Dcap U$ satisfying that ${nin{ mathbb Z}_+: T^nxin U, S^nyin V}$ is syndetic. Some characterization for the general case is also described. As applications we show that if a transitive system $(X,T)$ is disjoint from all minimal systems, then so are $(X^n,T^{(n)})$ and $(X, T^n)$ for any $nin { mathbb N}$. It turns out that a transitive system $(X,T)$ is disjoint from all minimal systems if and only if the hyperspace system $(K(X),T_K)$ is disjoint from all minimal systems.