#definition #forcing #forcing_property # Definition A subset $C\subseteq \mathbb{P}$ is called - $n$-linked if any $n$ elements of $C$ are compatible, i.e. have a common extension (perhaps not in $C$ itself). - linked if it is $2$-linked. - centered if it is $n$-lined for every $n$. I.e. every finite subset has a common extension. A poset is called - $\sigma$-centered if it is the union of $\omega$ many centered subsets. - $\sigma$-$n$-linked if it is the union of $\omega$ many $n$-linked subsets. - $\sigma$-linked if it is $\sigma$-2-linked. # Properties # Relations ![[Pasted image 20230215165614.png]] ## Implied by - [[]] ## Implies - [[Chain conditions|ccc]] - [[Knaster and pre-caliber]]