TITLE: Hausdorff spaces from continuous functions
QUESTION [5 upvotes]: I am trying to solve Exercise 2.35 in John M. Lee. Introduction to Topological Manifolds, p. 32:

Let $X$ be a topological space. Assume that for every $p\in X$ there exists a continuous function $f:~X\longrightarrow\mathbb{R}$ such that $f^{-1}(0)=\{p\}$. Show that $X$ is Hausdorff.

(The inverse $f^{-1}$ here is implied as converse, not a bijective inverse.)
My thinking is that if we take the open subset $(-1;1)$ of $\mathbb{R}$ for each $f$ since the map is continuous we get open sets containing $p_i$, and it boils down to showing the intersection is empty. I can't quite follow this important part through.

REPLY [2 votes]: For each continuous function $f\colon X\to \mathbb R$, define $f^{\times}$ on $X\times X$ by $f^{\times}(p,q)=f(p)-f(q)$. 
Consider the intersection: $$\bigcap_{f\colon X\to\mathbb R}\left(f^{\times}\right)^{-1}(\{0\})$$