$T$ is the set (say $S^\ast$) of all finite unions of finite intersections of elements of $S$.

Indeed, clearly $T\supseteq S^\ast$. It is also clear that $S^\ast$ is closed wrt to finite unions. Finally, $S^\ast$ is closed wrt to finite intersections, because 
$$\bigcap_{j=1}^n\bigcup_{i_j\in I_j}A_{i_j,j}
=\bigcup_{(i_1,\dots,i_n)\in I_1\times\cdots\times I_n}\,\bigcap_{j=1}^n A_{i_j,j}$$
for any sets $A_{i,j}$ and any index sets $I_j$, and finite intersections of finite intersections of sets are finite intersections of sets. 

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Similarly/dually, $T$ is the set of all finite intersections of finite unions of elements of $S$.

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Cf. [conversion to the conjunctive normal form and the disjunctive normal form][1].


  [1]: https://en.wikipedia.org/wiki/Conjunctive_normal_form#Conversion_to_CNF