(a) $\mathbb{R}$ . Using the hint and Theorem 22.2, or rather Corollary 22.3, $g$ is surjective and continuous (the preimage of an interval is the set of all points between two parabolas), and, by Corollary 22.3, it induces a bijective continuous map $f:X^{*}\rightarrow\mathbb{R}$ . Moreover, $g\circ h=i_{\mathbb{R}}$ where $h(x)=x\times0$ is a continuous (the preimage of $U\times V$ is either $\emptyset$ or $U$ ) right inverse of $g$ . So, by Exercise 2(a), $g$ is a quotient map, and, by Corollary 22.3, $f$ is a homeomorphism.

(b) $\overline{\mathbb{R}}_{+}$ . Similar to (a), using $g(x\times y)=x^{2}+y^{2}$ . $g$ is surjective and continuous (the preimage of $(a,b)$ for $0\le a<b$ is the set of all points between two circles, or punctured open disc if $a=0$ , and the preimage of $[0,c)$ is an open disc). Moreover, $g\circ h=i_{\overline{R}_{+}}$ where $h:\overline{\mathbb{R}}_{+}\rightarrow\mathbb{R}^{2}$ defined as $h(x)=\sqrt{x}\times0$ is continuous as the composite of two continuous functions $\sqrt{x}:\overline{\mathbb{R}}_{+}\rightarrow\overline{\mathbb{R}}_{+}$ (the preimage of $(a,b)$ for $0\le a<b$ is $(a^{2},b^{2})$ , and the preimage of $[0,c)$ is $[0,c^{2})$ ) and $x\times0:\overline{\mathbb{R}}_{+}\rightarrow\mathbb{R}^{2}$ (the preimage of $U\times V$ is either $\emptyset$ or $U\cap\overline{R}_{+}$ ). Hence, $g$ induces a homeomorphism of $X^{*}=\mathbb{R}^{2}/\sim$ with $\overline{\mathbb{R}}_{+}$ .