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Let $A \subset X$, $B \subset Y$ topological spaces and $A,B$ are proper subset. Prove that if $X,Y$ are connected, then $$X\times Y- A\times B $$ is connected.

Intuitively is clear but I don't know how to begin. Can anyone give me some advice to solve it?

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  • $\begingroup$ I missed something the subsets are proper subsets. $\endgroup$ – YTS Feb 14 '15 at 20:44
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Hint: Let $p\in X\setminus A\neq \emptyset$ and $q\in Y\setminus B\neq \emptyset$. For each $x\in X$, consider the subsets $T_x=\left(X\times \{q\}\right)\bigcup \left(\{x\}\times Y\right)$ and for each $y\in Y$, consider the subsets $S_y=\left(\{p\}\times Y\right)\bigcup \left(X\times \{y\}\right)$. Show that for all $x\in X$, $T_x$ are connected and for all $y\in Y$, $S_y$ are connected. Also note that $p\times q \in T_x\cap S_y$ for any $x\in X$ and $y\in Y$. Conclude that $$X\times Y\setminus A\times B=\left(X\times Y\setminus B\right)\cup \left(X\setminus A \times Y \right)=\bigcup\limits_{x\in X\setminus A} T_x\cup \bigcup\limits_{y\in Y\setminus B}S_y$$ is connected.

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