Let $\displaystyle \tau$ a stopping time w.r.t. the filtration $\displaystyle (F_n)_{n\in{\mathbb{N}}}$. The stopped $\displaystyle \sigma$-field $\displaystyle F_\tau$ associated with $\displaystyle \tau$ is defined to be

$\displaystyle F_\tau:={\{A\in{F}:A\cap{\{\omega:\tau(\omega)=n\} \in{F_n} \text{ for all }n\geq {0}}\}$.

I know this definition is equivalent to

$\displaystyle F_\tau:={\{A\in{F}:A\cap{\{\omega:\tau(\omega)\leq {n}\}\in{F_n} \text{ for all }n\geq {0}}\}$.

But is it the same as

$\displaystyle F_\tau:={\{A\in{F}:A\cap{\{\omega:\tau(\omega)<n\} \in{F_n} \text{ for all }n\geq {0}}\}$?

I have a feeling the answer is no, but I don't know how to start disproving it. Any help is appreciated! Thanks!