Section 18: Problem 10 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 $f:A\rightarrow B$ and $g:C\rightarrow D$ be continuous functions. Let us define a map $f\times g:A\times C\rightarrow B\times D$ by the equation $(f\times g)(a\times c)=f(a)\times g(c).$ Show that $f\times g$ is continuous.

Let $U\times V$ be a basis element for the product topology of $B\times D$ , then $(f\times g)^{-1}(U\times V)=f^{-1}(U)\times g^{-1}(V)$ . Since $f$ and $g$ are continuous, the inverse images on the right are open, and their product is open in $A\times C$ .

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Section 18: Problem 10 Solution

Section 18: Problem 10 Solution