In a space $X$, we define $x\sim y$ if there is no descomposition of $X$ into two disjoint open sets, one which contains $x$ and the other $y$ (the equivalence classes are called quasi-components).
The statement is:
If $x_1\sim x_2$ in $X$ and $y_1\sim y_2$ in $Y$ then $(x_1,y_1)\sim (x_2,y_2)$ in $X\times Y$. Or equivalently, $(x_1,y_1)\nsim (x_2,y_2)$ implies $x_1\nsim x_2$ or $y_1\nsim y_2$.
If $(x_1,y_1)\nsim (x_2,y_2)$, there would be $A,B$ open in $X\times Y$ such that $X\times Y=A\cup B, A\cap B= \emptyset$ and $(x_1,y_1)\in A$ and $(x_2,y_2)\in B$.
How can I show $x_1\nsim x_2$ or $y_1\nsim y_2$? Any hint?
Thank you.