Can we deduce that a finite topology $T$ satisfies Frankl's union-closed set conjecture?

Question

Let $X$ be a finite set and $T$ be a topology on $X$. Then $T$ is both union-closed and intersection-closed. Can we deduce that $T$ satisfies Frankl's union-closed set conjecture?

(We know that a complement of a union-closed set is an intersection-closed set and the union-closed set conjecture is equivalent to the intersection-closed set conjecture.)

How is this a set theory question? – Asaf Karagila May 14 '16 at 07:10 — Asaf Karagila, May 14 '16 at 07:10

Fedor Petrov · Answer 1 · 2016-05-14T14:22:26.723

11

Yes. Consider the minimal non-empty open set $A$. Each open set $B$ either contains $A$ (denote the family of such open sets by $P$) or does not intersect $A$ (the family of such open sets is denoted by $Q$). Then $B\rightarrow B\sqcup A$ is an injective map from $Q$ to $P$. Thus $|P|\geqslant |Q|$, i.e., at least half of open sets contain $A$.

edited May 14 '16 at 14:22

answered May 14 '16 at 07:59

Fedor Petrov

102,548

Dear Fedor, many thanks for your nice and brief answer. – Ahmadi May 16 '16 at 06:57

Can we deduce that a finite topology $T$ satisfies Frankl's union-closed set conjecture?

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