Let $\Omega$ be a set and $\{F_t\}_{t \geqslant 0}$ be a filtration. That is, it is a family of $\sigma$-algebras on $\Omega$ such that $F_{s} \subseteq F_t$ whenever $s \leqslant t$. Fix $t > 0$. Define :
\begin{equation*} F_{t+} = \bigcap_{\epsilon > 0} F_{t + \epsilon} \end{equation*} \begin{equation*} F_{t-} = \sigma (\bigcup_{s < t} F_{s}) \end{equation*}
And:
\begin{equation*} F_{t++} = \bigcap_{\epsilon > 0} F_{(t + \epsilon)+}= \bigcap_{\epsilon > 0} \bigcap_{\delta > 0} F_{t + \epsilon + \delta} = \bigcap_{\epsilon > 0} F_{t + \epsilon} = F_{t+} \end{equation*} \begin{equation*} F_{t+-} = \bigcap_{\epsilon > 0} F_{(t + \epsilon)-} = \bigcap_{\epsilon > 0} \sigma (\bigcup_{s < t + \epsilon} F_{s}) \end{equation*} \begin{equation*} F_{t-+} = \sigma (\bigcup_{s < t} F_{s+}) = \sigma (\bigcup_{s < t} \bigcap_{\epsilon > 0} F_{s+\epsilon}) \end{equation*} \begin{equation*} F_{t--} = \sigma (\bigcup_{s < t} F_{s-}) = \sigma (\bigcup_{s < t} \sigma(\bigcup_{u < s} F_{u})) \end{equation*}
The $F_{t++}$ simplifies really well. The other three not so much. Is there any simpler form one can obtain for the other three? In particular, do we get examples where (at least one of the following proper inclusion) :
\begin{equation*} F_{t--} \subset F_{t-} \subset F_{t-+} \subset F_t \subset F_{t+-} \subset F_{t+} \end{equation*}
