I'm having trouble with the following exercise from J.M Lee's "Intro to Topological Manifolds".
Suppose $X$ is a topoligical space, and for every $p \in X$ there exists a continuous $f:X \rightarrow \mathbb{R}$ such that $f^{-1}(0)=\{p\}$. Show that $X$ is Hausdorff.
Most all the other exercises require one or two lines so it should be v simple...
Obviously for any other $q \in X$, $f(q) \neq 0$ and so you have two disjoint open interavals $I_{0}, I_{f(q)}$ in $\Bbb R$ such that $0=f(p) \in I_0$ and $f(q) \in I_{f(q)}$. Since $f$ is continuous, $f^{-1}(I_{0})$ and $f^{-1}(I_{f(q)})$ are open in $X$ and contain $p$ and $q$ respectively. What I can't see is why they should be disjoint.
You have disjoint open intervals $I=I_0$ and $J=I_{f(q)}$. As you say, $f^{-1}(I)$ and $f^{-1}(J)$ are open neighbourhoods of $p$, $q$, respectively. Then $f^{-1}(I)\cap f^{-1}(J)=f^{-1}(I\cap J)=f^{-1}(\emptyset)=\emptyset$. Or, if $u\in f^{-1}(I)\cap f^{-1}(J)$ then $f(u)\in I$ and $f(u)\in J$ which is impossible as $I$ and $J$ are disjoint.