Section 30: Problem 14 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

Consider any covering of the product by basis open sets $U_{\alpha}\times V_{\alpha}$ . For every $x$ find a finite subcovering $U_{n}^{x}\times V_{n}^{x}$ that covers $x\times Y$ (the projection is an open map). Let $U^{x}=\cap_{n}U_{n}^{x}$ . Find a countable subcovering $U^{x_{k}}$ of $X$ . Then $\{U_{n}^{x_{k}}\times V_{n}^{x_{k}}\}$ is a countable subcovering of $X\times Y$ . Indeed, for any point $(x,y)$ there is $k$ such that $x\in U^{x_{k}}$ and also $n$ such that $(x_{k},y)\in U_{n}^{x_{k}}\times V_{n}^{x_{k}}$ . Since $U^{x_{k}}\subseteq U_{n}^{x_{k}}$ , we have $(x,y)\in U_{n}^{x_{k}}\times V_{n}^{x_{k}}$ .

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Section 30: Problem 14 Solution

Section 30: Problem 14 Solution