nLab upper set

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Definition

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

  • whenever xyx \leq y and xUx \in U, then yUy \in U.

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

A={y|x,xAxy}. 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 xx is the upper set generated by xx.

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

Last revised on June 9, 2022 at 19:43:53. See the history of this page for a list of all contributions to it.