Section 17: Problem 13 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

Show that $X$ is Hausdorff if and only if the diagonal $\Delta=\{x\times x|x\in X\}$ is closed in $X\times X$ .

Cool! $\bigtriangleup$ is closed in $X\times X$ iff for every $x\neq y$ there are two open sets $U$ containing $x$ and $V$ containing $y$ such that for no point $(z,z)\in U\times V$ iff any pair of different points have disjoint neighborhoods.

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Section 17: Problem 13 Solution

Section 17: Problem 13 Solution