I am sorry to ask this trivial question:
Let $R$ be a square $[0,1]\times[0,1]$, and $A$ is a continuous curve from $(0,0)$ to $(1,1)$, while $B$ is another continuous curve from $(0,1)$ to $(1,0)$. Show that $A$ and $B$ always intersect.
I have tried several methods, e.g., fixed point theorem and planar graph embedded on a Möbius band and etc. And I think this question is related to compactness and Hausdorff.