Section 18: 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

Given $x_{0}\in X$ and $y_{0}\in Y$ , show that the maps $f:X\rightarrow X\times Y$ and $g:Y\rightarrow X\times Y$ defined by $\begin{eqnarray*} f(x)=x\times y_{0} & \mbox{ and } & g(y)=x_{0}\times y \end{eqnarray*}$ are imbeddings.

Let $V=f(U)$ . Then $V$ is open in $X\times\{y_{0}\}$ iff $U\times\{y_{0}\}$ is open in $X\times\{y_{0}\}$ iff (using Exercise 4 of §16 one way, and by definition the other way) $U$ is open in $X$ . Similarly for $g$ .

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Section 18: Problem 4 Solution

Section 18: Problem 4 Solution