Section 5: Problem 2 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 $n>1$ there is bijective correspondence of $A_{1}\times\cdots\times A_{n}\mbox{ with }(A_{1}\times\cdots\times A_{n-1})\times A_{n}.$

(b) Given the indexed family $\{A_{1},A_{2},\ldots\}$ , let $B_{i}=A_{2i-1}\times A_{2i}$ for each positive integer $i$ . Show there is bijective correspondence of $A_{1}\times A_{2}\times\cdots$ with $B_{1}\times B_{2}\times\cdots$ .

(a) $f(a_{1},a_{2},\ldots,a_{n})=((a_{1},\ldots,a_{n-1}),a_{n})$ . It is clearly injective and surjective.

Note: by $(a_{1},...,a_{k})$ we, of course, mean the function from the index set $\{1,\ldots,k\}$ to $\cup_{i=1}^{k}A_{i}$ such that $x(i)=a_{i}$ where $a_{i}\in A_{i}$ , i.e. it is the specific element of the product $\prod_{i=1}^{k}A_{i}$ . So, the range of $f$ consists of the pair where the first element is a function itself, and the second one is an element of $A_{n}$ .

(b) $f(a_{1},a_{2},\ldots)=((a_{1},a_{2}),(a_{3},a_{4}),\ldots)$ . It is clearly injective and surjective. See the note above.

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Section 5: Problem 2 Solution

Section 5: Problem 2 Solution