HMMT 2014 U10

Find the number of nonempty sets $\mathcal{F}$ of subsets of the set $\{1, \ldots, 2014\}$ such that: (a) For any subsets $S_{1}, S_{2} \in \mathcal{F}, S_{1} \cap S_{2} \in \mathcal{F}$. (b) If $S \in \mathcal{F}, T \subseteq\{1, \ldots, 2014\}$, and $S \subseteq T$, then $T \in \mathcal{F}$.