Section 17: Problem 15 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 the $T_{1}$ axiom is equivalent to the condition that for each pair of points of $X$ , each has a neighborhood not containing the other.

$X$ is a $T_{1}$ -space iff $\forall x\in X$ , $\{x\}$ is closed iff $\forall x\in X$ , $X-\{x\}$ is open iff $\forall x\in X$ and $y\in X-\{x\}$ , there is an open set $U$ such that $y\in U\subset X-\{x\}$ .

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Section 17: Problem 15 Solution

Section 17: Problem 15 Solution