# 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]]