Section 5: Problem 4 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 $m,n\in\mathbb{Z}_{+}$ . Let $X\neq\emptyset$ .

(a) If $m\le n$ , find an injective map $f:X^{m}\rightarrow X^{n}$ .

(b) Find a bijective map $g:X^{m}\times X^{n}\rightarrow X^{m+n}$ .

(d) Find a bijective map $k:X^{n}\times X^{\omega}\rightarrow X^{\omega}$ .

(e) Find a bijective map $l:X^{\omega}\times X^{\omega}\rightarrow X^{\omega}$ .

(f) If $A\subset B$ , find an injective map $m:(A^{\omega})^{n}\rightarrow B^{\omega}$ .

Let $x\in X$ . Please also see the note to Exercise 2.

(a) $f(x_{1},...,x_{m})=(x_{1},...,x_{m},x,...,x)$ . It is clearly injective.

(b) $g((x_{1},\ldots,x_{m}),(y_{1},\ldots,y_{n}))=(x_{1},\ldots,x_{m},y_{1},\ldots,y_{n})$ . It is clearly injective and surjective.

(d) $k((x_{1},...,x_{n}),(y_{1},y_{2},...))=(x_{1},...,x_{n},y_{1},y_{2},\ldots)$ . It is clearly injective (if $x$ ’s or $y$ ’s are different, so is the image) and surjective.

(e) $l((x_{1},x_{2},...),(y_{1},y_{2},...))=(x_{1},y_{1},x_{2},y_{2},...)$ . It is clearly injective and surjective.

(f) $m((a_{11},a_{12},\ldots),\ldots,(a_{n1},a_{n2},\ldots))$ $=(a_{11},a_{21},\ldots,a_{n1},a_{12},a_{22},\ldots,a_{n2},\ldots)$ . It maps to $B^{\omega}$ as $a_{ij}\in A\subset B$ , and is clearly injective. It may be not be surjective (unlike (e)) if $A$ is a proper subset of $B$ .

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Section 5: Problem 4 Solution

Section 5: Problem 4 Solution