Exercise 8.2-b from Czes Kosniowski's A First Course In Algebraic Topology I'm reading the book Czes Kosniowski's A First Course In Algebraic Topology and i can't solve the exercise:

Let $\mathscr{F}$ be the topology on $\mathbb{R}$ defined by: $U \in \mathscr{F}$ if and only if for each $s \in U$ there is a $t>s$ such that $[s,t] \subseteq U$. Prove that $(\mathbb{R}, \mathscr{F})$ is Haudorff.

Someone can help me to solve this question or give me some hint? I guess the the proof follow by definition of Hausdorff space

HINT: Let $x,y\in\mathbb{R}$ with $x\neq y$. We can assume that $x<y$. Bearing in mind the definition of a Hausdorff space, what can we say about $[x,y)$ and $[y,\infty)$?
