I meet these two exercises:
Q1: let $A$ be a proper subset of $X$, and $B$ be a proper subset of $Y$. If $X$ and $Y$ are connected, show that $(X\times Y)\setminus(A\times B)$ is connected.
Q2: Let $Y\subset X$. Assume that $X$ and $Y$ be connected. Show that if $A$ and $B$ form a separation of $X\setminus Y$, then $Y\cup A$ and $Y\cup B$ are connected.
Here is my attempt for QN2
I think to prove it by contradiction, assume $Y\cup A$ and $Y\cup B$ are not connected
then for $P$ and $Q$ disjoint $Y\cup A=P\coprod Q$ and for $M$ and $N$ disjoint $Y\cup B=M\coprod N$
$(Y\cup A)\cup (Y\cup B)=(P\coprod Q)\cup (M\coprod N)$
The left side will give $X$,and the right side can be written as a disjoint union , this contradicts the fact that $X$ is connected, so $Y\cup A$ and $Y\cup B$ must be connected
I need help for Q1,