Let $(X,\leq)$ be a poset and let $Y$ be a subset of $X$. Suppose this collection of objets has the property that for each $x\in X$, there exists a $y\in Y$ such that $x\leq y$. Is there a name for this property?
Here's an example: let $X$ be the collection of open sets of some topological space (with the order being containement: $V\subseteq U$ if and only if $U\leq V$), and let $Y$ be any basis for the topology. Then the property holds.
Another example: let $X=\mathbb{R}$ and $Y=\mathbb{N}$. This is the Archimedean property.
Yet another example: let $X$ be the spectrum of some commutative ring with unity (where the ideals are partially ordered by inclusion) and let $Y$ be its maximal spectrum (the collection of maximal ideals).
The subset $Y$ is called cofinal in $X$. See en.wikipedia.org/wiki/Cofinal_(mathematics).