Section 18: Problem 11 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 $F:X\times Y\rightarrow Z$ . We say that $F$ is continuous in each variable separately if for each $y_{0}$ in $Y$ , me map $h:X\rightarrow Z$ defined by $h(x)=F(x\times y_{0})$ is continuous, and for each $x_{0}$ in $X$ , the map $k:Y\rightarrow Z$ defined by $k(y)=F(x_{0}\times y)$ is continuous. Show that if $F$ is continuous, then $F$ is continuous in each variable separately.

Let $x\in h^{-1}(A)$ where $A$ is open in $Z$ , then $F(x,y_{0})\in A$ and there is a basis neighborhood $U\times V$ of $x\times y_{0}$ in $X\times Y$ such that $U\times V\subset F^{-1}(A)$ . It follows that $h(U)\subset A$ , $x\in U\subset h^{-1}(A)$ , and, therefore, $h^{-1}(A)$ is open.

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Section 18: Problem 11 Solution

Section 18: Problem 11 Solution