Let $(X,T$) be a nontrivial dynamical system that is weakly mixing. Show that $X$ doesn't have isolated points.
Let's assume that $(X,T)$ has isolated point.
$(X,T)$ is weakly mixing, so $(X\times X, T\times T)$ is transitive. So $(X\times X, T\times T)$ contains a point with a dense orbit.
I am not sure how to use this information. My idea is to consider that point $(y_1,y_2) \in X\times X$ with a dense orbit, and let $x\in X$ be a isoleted point. So $\overline{O(y_1,y_2)}= X\times X$. So there is an $n_0 \in \mathbb{N}$ such that $dist((T\times T)^{n_0}(y_1,y_2),(x,x))<\epsilon$. But it would mean that $(T\times T)^{n_0}(y_1,y_2)=(x,x)$. However, we are far from a contradiction.
