In a poset or even proset, an upper set is a subset that is ‘upwards closed’; that is,
Upper sets form a Moore collection and so one can speak of the upper set generated by an arbitrary subset :
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 is the upper set generated by .
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.
Last revised on June 9, 2022 at 23:43:53. See the history of this page for a list of all contributions to it.