Section 16: Problem 3 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 the set $Y=[-1,1]$ as a subspace of $\mathbb{R}$ . Which of the following sets are open in $Y$ ? Which are open in $\mathbb{R}$ ? $\begin{eqnarray*} A & = & \{x|\tfrac{1}{2}<|x|<1\},\\ B & = & \{x|\tfrac{1}{2}<|x|\le1\},\\ C & = & \{x|\tfrac{1}{2}\le|x|<1\},\\ D & = & \{x|\tfrac{1}{2}\le|x|\le1\},\\ E & = & \{x|0<|x|<1\mbox{ and }1/x\notin\mathbb{Z}_{+}\}. \end{eqnarray*}$

$A$ and $E$ are open in $\mathbb{R}$ (as unions of open intervals), and, therefore, in $Y$ . $B$ is the only set open in $Y$ (as the intersection of a larger open set $\{x|\tfrac{1}{2}<|x|<2\}$ with $Y$ ), but not in $\mathbb{R}$ . $C$ and $D$ are not open in $Y$ , as their points $\pm\tfrac{1}{2}$ belong to their boundaries.

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Section 16: Problem 3 Solution

Section 16: Problem 3 Solution