If $X$ is a finite set, what is the smallest (in cardinality) family of open subsets $\mathcal U\subseteq 2^X$ such that $\mathcal U$ generates the discrete topology, i.e. if $\mathcal U\subseteq \tau\subseteq 2^X$ and $\tau$ is a topology, then $\tau=2^X$?



Smallest family of subsets that generates the discrete topology
Let $\mathcal{U}=\{A\_1,\ldots,A\_k\}$. Then for any element $x\in X$ there should exist a set $I(x)\subset \{1,\ldots,k\}$ such that $\cap\_{i\in I(x)} A\_i=\{x\}$. Note that $I(x)$ is not contained in $I(y)$ for $x\ne y$. Therefore $|X|\leqslant \binom{k}{\lfloor k/2\rfloor}$ by [Sperner's theorem](https://en.wikipedia.org/wiki/Sperner%27s_theorem). On the other hand, if $|X|\leqslant \binom{k}{\lfloor k/2\rfloor}$, we may construct an injection $f$ from $X$ to $\lfloor k/2\rfloor$-subsets of $\{1,\ldots,k\}$ and define $A\_i=\{x:i\in f(x)\}$. Then $\cap\_{i\in f(x)} A\_i=\{x\}$.


So the answer is the minimal $k$ for which $|X|\leqslant \binom{k}{\lfloor k/2\rfloor}$.


