Section 4: Problem 3 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

(a) Show that if $\mathcal{A}$ is a collection of inductive sets, then the intersection of the elements of $\mathcal{A}$ is an inductive set.

(b) Prove the basic properties (1) and (2) of $\mathbb{Z}_{+}$ .

(a) For every $x\in\cap_{A\in\mathcal{A}}A$ , $x\in A$ for all $A\in\mathcal{A}$ , therefore, $x+1\in A$ for all $A\in\mathcal{A}$ , therefore, $x+1\in\cap_{A\in\mathcal{A}}A$ . Moreover, $1\in A$ for all $A\in\mathcal{A}$ , hence, $1\in\cap_{A\in\mathcal{A}}A$ , and $\cap_{A\in\mathcal{A}}A$ is inductive.

(b) $\mathbb{Z}_{+}$ is inductive by (a). And if $A$ is an inductive set of positive integers, then it is both a subset and superset of $\mathbb{Z}_{+}$ .

Subpages

Section 4: Problem 3 Solution

Section 4: Problem 3 Solution