Frankl's conjecture restricted to finite topological spaces

A finite topological space is a finite family of finite sets that is closed under both union and intersection.

Frankl's 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?

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 $|\mathcal F_1|\geq|\mathcal F_0|,|\mathcal F_1|\geq|\mathcal F|/2$.
• You are asking whether Frankl's conjecture holds for distributive lattices. This was stated by Rival and proved by Poonen and subsequently generalized in various ways, e.g., lower semimodular lattices. See zaik.uni-koeln.de/~schaudt/UCSurvey.pdf. It is also easy to prove for lattices such that for every $\hat{0}<s\leq t$ there exists $u\neq t$ for which $s\vee u=t$. See Enumerative Combinatorics, vol. 1, 2nd ed., Exercise 3.102(b). Nov 26 '17 at 21:22