Question 49
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 $T \in \mathcal{F}$.
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 $T \in \mathcal{F}$.