I'm reading topology of Munkres and I have a problem that stuck me for a while. I'm so greatful if anyone can help me with this.
Let $A$ be a proper subset of $X$, and let $B$ is a proper subset of $Y$. If $X$ and $Y$ are connected, show that $$(X\times Y) \setminus (A\times B)$$ is connected.
Thanks so much for your consideration ^^
Best Answer
We can simplify Davide Giraudo's answer by noting that we only need to show that $(a,b)$ is in the same connected component as every other point.
So, start by fixing $a \in X \setminus A$ and $b \in Y \setminus B$ as Davide does, and consider an arbitrary point $(x,y) \in (X \times Y) \setminus (A \times B)$.
If $x \notin A$, then $\{x\} \times Y$ is connected and contains both $(x,y)$ and $(x,b)$, while $X \times \{b\}$ is connected and contains both $(x,b)$ and $(a,b)$. Thus, $(\{x\} \times Y) \cup (X \times \{b\})$ is connected and contains both $(x,y)$ and $(a,b)$.
Otherwise, $x \in A \implies y \notin B$. Thus, analogously, $X \times \{y\}$ is connected and contains both $(x,y)$ and $(a,y)$, while $\{a\} \times Y$ is connected and contains both $(a,y)$ and $(a,b)$, and so $(X \times \{y\}) \cup (\{a\} \times Y)$ is connected and contains both $(x,y)$ and $(a,b)$.