Can someone reconcile the two definition of Suslin's condition?

Question

I am given two definitions of the so called "Suslin's condition" and I need to reconcile them. I am an undergrad and this is just for exploration.

Definition 1.

A partially ordered set $X$ is said to satisfy the Suslin's condition (or countable chain condition), if every strong antichain in $X$ is countable.

Definition 2:

Given some topological space $(X, \tau)$, $(X, \tau)$ satisfies the Suslin's condition if there are no uncountable collection of mutually disjoint open subset of $X$

These definitions looks very very different to me, can someone please elaborate on how every strong antichain in $X$ is countable means no uncountable collection of mutually disjoint open subset of $X$

Thanks!

Answer 1

The topological condition is a special case of the partial order one.

If we have $(X,\tau)$, we have a partial order $(\tau\setminus\{\emptyset\}, \subseteq)$, the non-empty open sets, ordered by inclusion.

Two open sets are disjoint iff they have no common lower bound in the p.o., because the only candidate would be the empty set (the intersection of two open sets is always open; if non-empty it would be a common lower bound).

So a pairwise disjoint family of non-empty open sets is precisely a strong antichain in this partial order, and Suslin's condition in partial orders says all those strong antichains are at most countable (or equivalently there is no such uncountable collection), precisely the Suslin condition on topological spaces.