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 $(A,\preceq)$ is a subset, $U\subseteq A$ with the following property:

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

See also

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