Upper section

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This may sometimes be called an upper set, according to wikipedia, however I am dealing with presets not posets here, so upper set might only be for posets, and upper section for presets, or both. Not sure needs investigation

Definition

An upper section of a preset [ilmath](A,\preceq)[/ilmath] is a subset, [ilmath]U\subseteq A[/ilmath] with the following property:

• [ilmath]\forall a,b\in A[(a\in U\wedge a\preceq b)\implies b\in U][/ilmath], that is:
  • Given an [ilmath]a[/ilmath] and [ilmath]b[/ilmath] in [ilmath]A[/ilmath], if [ilmath]a[/ilmath] is in [ilmath]U[/ilmath] and [ilmath]a\preceq b[/ilmath] then we require [ilmath]b\in U[/ilmath]

See also

• Lower section - the dual concept to this
• [ilmath]\Uparrow[/ilmath]-functor, a functor, [ilmath]:[/ilmath][ilmath]\mathrm{PRE} [/ilmath][ilmath]\leadsto[/ilmath][ilmath]\mathrm{TOP} [/ilmath] that uses this notion.
  • Closely related to the Alexandroff topology