example of this connected set in $[0,1]\times[0,1]$ How to find an example that satisfies the following conditions?
$A,B$ are connected subsets of $[0,1]\times[0,1]$ such that $\:(0,0),(1,1) \in A$, $\:(0,1),(1,0) \in B$, $\:A \cap B$ is empty.
How can I find this?? Help me…
 A: What you want cannot be achieved if you replace “connected” by “path-connected”.
However, you can do the trick with the “topologist's sine curve” $S=\{(x,\sin\frac1x):x\in(0,1]\}$.
Scale this graph and put it somewhere in the interior of the square, calling the resulting set $A$. Shift $A$ a little bit below and call the resulting set $B$. This “little bit” should be so small that the vertical closed line segment $\overline A\setminus A$ and the shifted version $\overline B\setminus B$ intersect in some nondegenerate (vertical closed) line segment $C$.
It remains to complete the picture: Draw straight lines $L_1$ and $L_2$ from $(0,0)$ and $(0,1)$ to the lower and upper ends of $C$, respectively. And extend the curve from $A$ and $B$ by disjoint “smooth” paths $P_1$ and $P_2$ to the corners $(1,1)$, and $(1,0)$, respectively.
Then the two sets you look for are $P_1\cup A\cup L_1$ and $P_2\cup B\cup L_2$, respectively: It is clear that the unions $P_1\cup A$ and $P_2\cup B$ are connected. And the fact that $A\cup L_1$ and $B\cup L_2$ are connected follows in the same way as you can prove that any set $M$ with $S\subseteq M\subseteq\overline S$ is connected.
