Section 21: 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

Using the closed set formulation of continuity (Theorem 18.1), show that the following are closed subsets of $\mathbb{R}^{2}$ : $\begin{eqnarray*} A & = & \{x\times y|xy=1\},\\ S^{1} & = & \{x\times y|x^{2}+y^{2}=1\},\\ B^{2} & = & \{x\times y|x^{2}+y^{2}\le1\}. \end{eqnarray*}$ The set $B^{2}$ is called the (closed) unit ball in $\mathbb{R}^{2}$ .

Using Lemma 21.4, the sets are continuous preimages of closed subsets of $\mathbb{R}$ , and we may apply Theorem 18.1.

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Section 21: Problem 10 Solution

Section 21: Problem 10 Solution