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

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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.

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