Proposition 5

Let $X$ be a topological space and suppose that for each pair of distinct points $x, y$ of $X$ there is a continuous mapping $f$ of $X$ into a Hausdorff space $X'$ such that $f(x) \neq f(y)$. Then $X$ is Hausdorff.

Let $V'$ and $W'$ be disjoint neighbourhoods of $f(x)$ and $f(y)$ respectively in $X'$; then $\overline{f}^{-1}(V')$ and $\overline{f}^{-1}(W')$ are disjoint neighbourhoods of $x$ and $y$ respectively in $X$.