(a) Define the subspace, quotient and product topologies.

(b) Let $X$ be a compact topological space and $Y$ a Hausdorff topological space. Prove that a continuous bijection $f: X \rightarrow Y$ is a homeomorphism.

(c) Let $S=[0,1] \times[0,1]$, equipped with the product topology. Let $\sim$ be the smallest equivalence relation on $S$ such that $(s, 0) \sim(s, 1)$ and $(0, t) \sim(1, t)$, for all $s, t \in[0,1]$. Let

$$T=\left\{(x, y, z) \in \mathbb{R}^{3} \mid\left(\sqrt{x^{2}+y^{2}}-2\right)^{2}+z^{2}=1\right\}$$

equipped with the subspace topology from $\mathbb{R}^{3}$. Prove that $S / \sim$ and $T$ are homeomorphic.

[You may assume without proof that $S$ is compact.]