Section 1: Problem 5 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 nonempty collection of sets. Determine the truth of each of the following statements and of their converses:

(a) $x\in\cup_{A\in\mathcal{A}}A$ $\Rightarrow$ $x\in A$ for at least one $A\in\mathcal{A}$ .

(b) $x\in\cup_{A\in\mathcal{A}}A$ $\Rightarrow$ $x\in A$ for every $A\in\mathcal{A}$ .

(d) $x\in\cap_{A\in\mathcal{A}}A$ $\Rightarrow$ $x\in A$ for every $A\in\mathcal{A}$ .

(a) and (d): $\Leftrightarrow$ (by definition). (b): $\Leftarrow$ . (c): $\Rightarrow$ .

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Section 1: Problem 5 Solution

Section 1: Problem 5 Solution