Let $X$ and $Y$ be compact, connected and metric spaces (continuum), $p_X\colon X\times Y\to X$ the projection in $X$ and $p_Y\colon X\times Y\to Y$ the projection in $Y$. Suppose that $K$ and $L$ are subcontinua of $X\times Y$ such that the following conditions holds:
- $p_X(K)=p_X(L)=X$,
- $p_Y(K)=p_Y(L)=Y$.
That is, $K$ and $L$ have full projections. Is it true that $K\cap L\neq \varnothing$?
This is not true, eg consider the threads on a screw. More specifically if $X=[0,1]$, and $Y=S^1$, the subspaces $X_1$ and $X_2$ given by
$$X_1=\left\{\left(t,e^{2\pi it}\right):0\leq t\leq 1\right\}$$ $$X_2=\left\{\left(t,e^{2\pi i(t+0.5)}\right):0\leq t\leq 1\right\}$$ both have full projections and are closed and connected, but do not intersect.