nLab upper set

Definition

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

• whenever $x \leq y$ 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 $A$:

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

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 $x$.

An inhabited, open upper set of rational numbers (or equivalently of real numbers) determines precisely an upper real number.

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

See also

• posite
• lower set
• ideal
• filter
• copresheaf
• presheaf