= Solution
Put $D=\{(\omega,t):\omega\geq t\}$ and let $\mathcal C$ denote the proposed family of <sets>. This is a <sigma-algebra>: complements replace $T$ and $A$ by their complements, and countable unions replace them by their unions. For a generating <predictable rectangle> $B\times(s,u]$, part (a) says either $B\subseteq(0,s)$ or $B^c\subseteq(0,s)$. In the first case its trace on $D$ is empty; in the second it is exactly $D\cap(\Omega\times(s,u])$. Its trace on $D^c$ is always product-Borel. Hence $\mathcal P\subseteq\mathcal C$.
For the reverse inclusion, $X$ is left-continuous and adapted, so $D=\{X=1\}$ belongs to the <predictable sigma-algebra>. Every deterministic <Borel set> in time also belongs to that <sigma-algebra>, hence $D\cap(\Omega\times T)$ is predictable. For a product rectangle $E\times T$,
$$
(E\times T)\cap D^c=\bigcup_{s\in\mathbb Q_{>0}}\bigl((E\cap(0,s))\times(s,\infty)\bigr)\cap(\Omega\times T).
$$
Each term is predictable because $E\cap(0,s)\in\mathcal F_s$. Every pair with $\omega<t$ lies in such a term by choosing $\omega<s<t$. The class of product-Borel <sets> $A$ for which $A\cap D^c$ is predictable is a <sigma-algebra> containing the product rectangles, so it contains $\mathcal F\otimes\mathcal F$. Thus
$$
\boxed{\mathcal P=\{(D\cap(\Omega\times T))\cup(D^c\cap A):T\in\mathcal F,\ A\in\mathcal F\otimes\mathcal F\}.}
$$
This <survival-observation predictable sigma-algebra> expresses a sharp distinction: before and at the lifetime the only observable coordinate is time, whereas strictly after it the lifetime is known.
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