my question is: If we have a poisson process $X=(X_t)_{t\in \mathbb R_+}$ and we denote by $D$ the nonnegative dyadic numbers. Then why does
\begin{align} P\Bigg[\bigcap_{t\in D} \bigcup\limits_{s\in D, s>t} \{ X_s=X_t \}\Bigg]=1 \end{align}
hold.
Can we use it as a limsup with Borel-Cantelli although we are dealing with dyadic numbers?
Maybe someone can help me.