Remark 2

For a topological space X to be separated (TG, I, p. 52, def. 1), it is necessary and sufficient that the mapping of X into a space reduced to a point be separated. In this case, every continuous mapping of X into a topological space is separated (prop. 1, (iii)).

Let $f: X\rightarrow$ Y be a continuous and separated mapping. For every point $y$ of Y, the fibre $\overset{-1}{f}(y)$ is a separated topological space (loc. cit.). There exist nevertheless continuous mappings which are not separated but all of whose fibres are separated topological spaces (I, p. 140, exerc. 1).