nLab
upper set

In a poset or even proset, an upper set $U$ is a subset that is ‘upwards closed’; that is,

• whenever $x\le y$ and $x\in U$, then $y\in U$.

Upper sets form a Moore collection and so one can speak of the upper set generated by an arbitrary subset $A$:

$$A\uparrow =\{y\mid \exists x,x\in A\wedge x\le y\}.$$A{\uparrow} = \{ y \;|\; \exists x,\; x \in A \;\wedge\; x \leq y \} .

Sometimes an upper set is called a ‘filter’, but that term can also mean something more specific (and always does in a lattice).

An upper set is also sometimes called an ‘up set’, but that term can also mean something more specific: the up set of $x$ is the upper set generated by $x$.

The upper sets form a topological structure on (the underlying set of) the proset, called the Alexandroff topology.