Section 20: Problem 6 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 $\overline{\rho}$ be the uniform metric on $\mathbb{R}^{\omega}$ . Given $\mathbf{x}=(x_{1},x_{2},\ldots)\in\mathbb{R}^{\omega}$ and given $0<\epsilon<1$ , let $U(\mathbf{x},\epsilon)=(x_{1}-\epsilon,x_{1}+\epsilon)\times\cdots\times(x_{n}-\epsilon,x_{n}+\epsilon)\times\cdots.$

(a) Show that $U(\mathbf{x},\epsilon)$ is not equal to the $\epsilon$ -ball $B_{\overline{\rho}}(\mathbf{x},\epsilon)$ .

(b) Show that $U(\mathbf{x},\epsilon)$ is not even open in the uniform topology.

(a) $\mathbf{y}=(x_{1}+\tfrac{\epsilon}{2},x_{2}+\tfrac{2\epsilon}{3},x_{3}+\tfrac{3\epsilon}{4},...)$ is in $U(\mathbf{x},\epsilon)$ but not in $B_{\overline{\rho}}(\mathbf{x},\epsilon)$ . But, of course, $B_{\overline{\rho}}(\mathbf{x},\epsilon)\subset U(\mathbf{x},\epsilon)$ . So that $B_{\overline{\rho}}(\mathbf{x},\epsilon)\subsetneq U(\mathbf{x},\epsilon)$ .

(b) For the point $\mathbf{y}$ from (a) there is no ball centered at it and contained in $U(\mathbf{x},\epsilon)$ .

(c) Clearly, for $\delta<\epsilon$ , $U(\mathbf{x},\delta)\subset B_{\overline{\rho}}(\mathbf{x},\epsilon)$ : the distance between any point in $U(\mathbf{x},\delta)$ and $\mathbf{x}$ is less or equal to $\delta<\epsilon$ . Hence, $\bigcup_{\delta<\epsilon}U(\mathbf{x},\delta)\subset B_{\overline{\rho}}(\mathbf{x},\epsilon)$ . Further, for $\mathbf{y}\in B_{\overline{\rho}}(\mathbf{x},\epsilon)$ , $\overline{\rho}(\mathbf{x},\mathbf{y})<\epsilon$ , and, therefore, $\mathbf{y}\in U(\mathbf{x},\tfrac{\epsilon+\overline{\rho}(\mathbf{x},\mathbf{y})}{2})$ where $\tfrac{\epsilon+\overline{\rho}(\mathbf{x},\mathbf{y})}{2}<\epsilon$ . Hence, $B_{\overline{\rho}}(\mathbf{x},\epsilon)\subset\bigcup_{\delta<\epsilon}U(\mathbf{x},\delta)$ .

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Section 20: Problem 6 Solution

Section 20: Problem 6 Solution