A [finite topological space](https://en.wikipedia.org/wiki/Finite_topological_space) is a finite family of finite sets that is closed under both union and intersection. 


[Frankl's conjecture](https://en.wikipedia.org/wiki/Union-closed_sets_conjecture) states that for any finite union-closed family of finite sets, other than the family consisting only of the empty set, there exists an element that belongs to at least half of the sets in the family.


Is Frankl's conjecture known to be true when restricted to finite topological spaces?



Frankl's conjecture restricted to finite topological spaces
Consider the smallest nonempty set $S$ in our family $\mathcal F$ and pick any $s\in S$. Let $\mathcal F\_0$ be the subfamily of sets not containing $s$ (including $\varnothing$) and $\mathcal F\_1$ the subfamily of sets containing $s$.


If $s\not\in A\in\mathcal F$, then $S\cap A$ is a smaller element of $\mathcal F$, so it must be empty. Hence the map $\mathcal F\_0\to\mathcal F\_1,A\mapsto S\cup A$ is an injection (with inverse $B\mapsto B\setminus S$). Hence $|\mathcal F\_1|\geq|\mathcal F\_0|,|\mathcal F\_1|\geq|\mathcal F|/2$.


