Section 17: 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 $A\subset X$ and $B\subset Y$ . Show that in the space $X\times Y$ , $\overline{A\times B}=\overline{A}\times\overline{B}.$

$(x,y)\in\overline{A\times B}$ iff for every basis element $U\times V$ of $X\times Y$ , where $U$ is open in $X$ and $V$ is open in $Y$ , such that $(x,y)\in U\times V$ the intersection $U\times V\cap A\times B\neq\emptyset$ iff for every neighborhood $U$ of $x$ and every neighborhood $V$ of $y$ , $U\cap A\neq\emptyset\neq V\cap B$ iff $x\in\overline{A}$ and $y\in\overline{B}$ iff $(x,y)\in\overline{A}\times\overline{B}$ .

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Section 17: Problem 9 Solution

Section 17: Problem 9 Solution