Section 23: Problem 9 Solution

Working problems is a crucial part of learning mathematics. No one can learn topology merely by poring over the definitions, theorems, and examples that are worked out in the text. One must work part of it out for oneself. To provide that opportunity is the purpose of the exercises.

James R. Munkres

Let $A$ be a proper subset of $X$ , and let $B$ be a proper subset of $Y$ . If $X$ and $Y$ are connected, show that $(X\times Y)-(A\times B)$ is connected.

Let $M=X\times Y-A\times B$ . For any $x\notin A$ , $V_{x}=\cup_{y\in Y}\{(x,y)\}\subset M$ is connected as a set homeomorphic to $Y$ . And for all $y\notin B$ , $H_{y}=\cup_{x\in X}\{(x,y)\}\subset M$ is connected as well. Suppose, $a\notin A$ and $b\notin B$ . Then, $C=V_{a}\cup H_{b}\subset M$ is connected (the two sets have the point $(a,b)$ in common). Moreover, for every $x\notin A$ and $y\notin B$ , $V_{x}$ and $H_{y}$ intersect $C$ , and $M=\cup_{x\notin A}V_{x}\cup\cup_{y\notin B}H_{y}$ . Therefore, by Exercise 3, $M$ is connected.

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Section 23: Problem 9 Solution

Section 23: Problem 9 Solution