Exercise 1, § 2

Let $X$ be a topological space, $Y = X\times \{0,1\}$ and let $R$ be an equivalence relation on $Y$ compatible with its natural structure of X-space. We denote by $f: Y/R\rightarrow X$ the canonical mapping. Let A be the set of the $x\in X$ such that $(x,0)$ and $(x,1)$ are equivalent.

a) Now prove that $f$ is separated if and only if A is closed in X.

b) Now prove that all the fibres of $f$ are separated topological spaces (cf. I, p. 26, remark 2).