# Show that $X$ is Hausdorff.

Suppose that $X$ is a space with the property that for any point $p \in X$ there is a map $f: X \rightarrow \mathbb{R}$ such that $f^{-1}(1) = \{p\}$. Show that $X$ is Hausdorff.

Solution:

Suppose that $x \neq y$ are points in $X$. Choose $f: X \rightarrow \mathbb{R}$ with $f(x) = 1$ and $f(y) \neq 1$ and open disjoint intervals around $1$ and $f(y)$ in $\mathbb{R}$ (which is possible since $\mathbb{R}$ is Hausdorff).

Now we are done if we can say that the inverse images of these intervals under $f$ are open disjoint sets in $X$ containing $x$ and $y$, respectively. Because then $X$ is Hausdorff.

$f$ is a map, i.e. a continuous function, and therefore its inverse takes open sets to open sets.

This is probably a very simple question and bordering on stupid for you experienced topologists but i just wonder how we can be sure that the inverse takes disjoint open sets to disjoint open sets?

Let $f\colon X\to Y$ be any map between sets, $A, B\subseteq Y$, $A\cap B=\emptyset$. Suppose there is an element $x\in f^{-1}(A)\cap f^{-1}(B)$. Then we have $f(x)\in A\cap B=\emptyset$ a contradiction.