Let  $\tau$ be a random variable, which is defined on the filtered probability space $(\Omega, \mathcal{F}, (\mathcal{F})_{t\in T}, P)$ with values in $T$. In most cases, $T=[0,\infty]$. Then  $\tau$ is called a stopping time (with respect to the filtration $(\mathcal{F})_{t\in T}$), if the following condition holds:
$\{\tau\leq t\}\in (\mathcal{F})_t $ for all  $t\in T$.  Can someone explain a little bit what is ? 
Can we replace $\{\tau\leq t\}\in (\mathcal{F})_t $  using $\{\tau<t\}\in (\mathcal{F})_t $ in the definition of stopping time?