You probably don’t need $* \in \{\cap, \cup\}$ if you only use it for $\cap$. The union for (O2) shouldn’t be over the product $I_{j} \times J$, since $I_{j}$ depends on $J$. Instead, it should be something like: Let $\{U_{j}\}_{j \in J}\subset \tau(\mathcal{B})$. For each $j \in J$, let $I_{j}$ such that $U_{j} = \bigcup_{i \in I_j}B_{i, j}$, then
You probably don’t need $* \in \{\cap, \cup\}$ if you only use it for $\cap$. The union for (O2) shouldn’t be over the product $I_{j} \times J$, since $I_{j}$ depends on $J$. Instead, it should be something like: Let $\{U_{j}\}_{j \in J}\subset \tau(\mathcal{B})$. For each $j \in J$, let $I_{j}$ such that $U_{j} = \bigcup_{i \in I_j}B_{i, j}$, then