Section 13: Problem 1 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 $X$ be a topological space; let $A$ be a subset of $X$ . Suppose that for each $x\in A$ there is an open set $U$ containing $x$ such that $U\subset A$ . Show that $A$ is open in $X$ .

For every $x$ there is an open set $U_{x}$ such that $x\in U_{x}\subset A$ , therefore, $U=\cup_{x\in A}U_{x}$ is open, and $A=\cup_{x\in A}\{x\}$ $\subset\cup_{x\in A}U_{x}$ $\subset A$ , i.e. $U=A$ .

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Section 13: Problem 1 Solution

Section 13: Problem 1 Solution