Section 22*: 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

Let $\pi_{1}:\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}$ be projection on the first coordinate. Let $A$ be the subspace of $\mathbb{R}\times\mathbb{R}$ consisting of all points $x\times y$ for which either $x\ge0$ or $y=0$ (or both); let $q:A\rightarrow\mathbb{R}$ be obtained by restricting $\pi_{1}$ . Show that $q$ is a quotient map that is neither open nor closed.

Consider $f:A\rightarrow\mathbb{R}\times\{0\}$ defined as $f(x,y)=(x,0)$ . Then, $f$ is a retraction (as a continuous function on a restricted domain), hence, it is a quotient map (Exercise 2(b)). There is an obvious homeomorphism $h$ of $\mathbb{R}\times\{0\}$ with $\mathbb{R}$ defined by $h(x,0)=x$ (see also Exercise 4 of §18). Moreover, $q=h\circ f$ . Therefore, $q$ is a quotient map as well (Theorem 22.2). But $q(\mathbb{R}\times\mathbb{R}_{+}\cap A)=\overline{\mathbb{R}}_{+}$ is not open in $\mathbb{R}$ , and $q(\{(x,1/x)|x>0\})=\mathbb{R}_{+}$ is not closed in $\mathbb{R}$ .

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Section 22*: Problem 3 Solution

Section 22*: Problem 3 Solution