(a) Let $X$ be a topological space. Define what is meant by a quotient of $X$ and describe the quotient topology on the quotient space. Give an example in which $X$ is Hausdorff but the quotient space is not Hausdorff.

(b) Let $T^{2}$ be the 2-dimensional torus considered as the quotient $\mathbb{R}^{2} / \mathbb{Z}^{2}$, and let $\pi: \mathbb{R}^{2} \rightarrow T^{2}$ be the quotient map.

(i) Let $B(u, r)$ be the open ball in $\mathbb{R}^{2}$ with centre $u$ and radius $r<1 / 2$. Show that $U=\pi(B(u, r))$ is an open subset of $T^{2}$ and show that $\pi^{-1}(U)$ has infinitely many connected components. Show each connected component is homeomorphic to $B(u, r)$.

(ii) Let $\alpha \in \mathbb{R}$ be an irrational number and let $L \subset \mathbb{R}^{2}$ be the line given by the equation $y=\alpha x$. Show that $\pi(L)$ is dense in $T^{2}$ but $\pi(L) \neq T^{2}$.