please help me with this topology problem:
A space is said to be contractible if it is homotopy equivalent to a singleton. Say whether each of the following statements is true or false and justify:
(i) Any subspace of a contractible space is contractible.
(ii) Any quotient of a contractible space is contractible.
(iii) The product of two contractible spaces is contractible.
My thoughts：
i) False. Consider X = $R^2$ and X = $S^1 \subseteq X$, then X is contractible because $R^2$ is convex and any convex space is contractible. Also, we know that $\pi_1 (S^1) = \mathbb{Z}$ and for contractible space Z we have $\pi_1 (Z) = {0}$. Thus, $S^1$ is not contractible.
ii) False. Consider X = $\mathbb{R}$ and Y = $\mathbb{Z}$, then $$X/Y = \mathbb{R} / \mathbb{Z} \cong S^1.$$ Now $\mathbb{R}$ is contractible as it's convex, but $S^1$ being non contractible with $\pi_1 (S^1) = \mathbb{Z}$ and $$\pi_1 (x/y) \cong \pi_1 (S^1) = \mathbb{Z} \, so \, \pi_1 (x/y) \neq \{0\},$$ so X/Y is not contractible.
iii) By contractibility of $X$ we have a homotopy $f_t:X\times I\to X$ from $id_X$ to $c_X=x_0$. Similarly for $Y$ we have a homotopy $g_t:Y\times I\to Y$ from $id_Y$ to $c_Y=y_0$.
Consider the map $h_t:X\times Y\times I\to X\times Y$ given by $(x,y,t)\mapsto (f_t(x),g_t(y))$. Clearly $h_t$ is continuous, and at $t=0$, $h_0(x,y)=(f_0(x),g_0(y))=(x,y)$, while at $t=1$, $h_1(x,y)=(f_1(x),g_1(y))=(x_0,y_0)$. Hence $h_t$ is a homotopy between $id_{X\times Y}$ and $c_{X\times Y}=(x_0,y_0)$.
Are my answers correct? If not, please point out the error. Thank you so much!