Is there a name for a partial order $\preceq$ on a set $X$ with the following property: "there exists a countable set $S \subset X$ such that for all $x \in X$ there exists $y \in S$ with $x \preceq y$"?


(**Remark:** If such a set $S$ exists, then it is possible to choose $S$ to be a chain: letting $(x\_n)$ be an enumeration of the original (not necessarily totally ordered) set $S$, just define $\tilde{x}\_1:=x\_1$ and then recursively choose $\tilde{x}\_n$ to be a point in $S$ that is greater than or equal to both $x\_n$ and $\tilde{x}\_{n-1}$.)



Is there a name for a partial order in which there is a countable chain which "dominates" the whole space?
Such a partial order would be said to have **countable [cofinality](https://en.wikipedia.org/wiki/Cofinality).** 


(The set $S$ would be said to be [cofinal](https://en.wikipedia.org/wiki/Cofinal_%28mathematics%29)).


(Strictly speaking, your property could also apply to an order with *finite* cofinality, so if that is a possibility, then depending on what the word "countable" means to you, you might want to say "cofinality at most countable.")


