Section 19: Problem 5 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

One of the implications stated in Theorem 19.6 holds for the box topology. Which one?

If $f:A\rightarrow\prod_{\alpha\in J}X_{\alpha}$ is continuous, and $\prod_{\alpha\in J}X_{\alpha}$ is in the box topology, then, since the product topology is coarser than the box topology, $f$ is still continuous if $\prod_{\alpha\in J}X_{\alpha}$ is given the product topology, and, by Theorem 19.6, each $f_{\alpha}$ , $\alpha\in J$ , is continuous.

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Section 19: Problem 5 Solution

Section 19: Problem 5 Solution