Section 9: Problem 7 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 $A$ and $B$ be two nonempty sets. If there is an injection of $B$ into $A$ , but no injection of $A$ into $B$ , we say that $A$ has greater cardinality than $B$ .

(a) Conclude from Theorem 9.1 that every uncountable set has greater cardinality than $\mathbb{Z}_{+}$ .

(b) Show that if $A$ has greater cardinality than $B$ , and $B$ has greater cardinality than $C$ , then $A$ has greater cardinality than $C$ .

(c) Find a sequence $A_{1},A_{2},\ldots$ of infinite sets, such that for each $n\in\mathbb{Z}_{+}$ , the set $A_{n+1}$ has greater cardinality than $A_{n}$ .

(d) Find a set that for every $n$ has cardinality greater than $A_{n}$ .

(a) An uncountable set is infinite (injection one way by Theorem 9.1; remember how the injection was constructed "by induction" in the proof of the theorem and then the role of the Axiom of Choice was discussed after the proof?), but not countably infinite (no injection the other way by Theorem 7.1).

(b) There are injections $C\rightarrow B\rightarrow A$ , therefore, there is an injection $C\rightarrow A$ . If there were an injection $A\rightarrow C$ then there would be an injection $B\rightarrow A\rightarrow C$ .

(d) $\mathcal{P}(\cup_{i}A_{i})$ .

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Section 9: Problem 7 Solution

Section 9: Problem 7 Solution