View source for Upper section

{{Stub page|Needs fleshing out}}
{{Requires references|This may sometimes be called an upper set, according to wikipedia, however I am dealing with [[preset|presets]] not [[poset|posets]] here, so upper set might only be for posets, and upper section for presets, or both. Not sure needs investigation}}
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==Definition==
An ''upper section'' of a [[preset]] {{M|(A,\preceq)}} is a subset, {{M|U\subseteq A}} with the following property{{rAITCTHS2010}}:
* {{M|\forall a,b\in A[(a\in U\wedge a\preceq b)\implies b\in U]}}, that is:
** Given an {{M|a}} and {{M|b}} in {{M|A}}, if {{M|a}} is in {{M|U}} and {{M|a\preceq b}} then we require {{M|b\in U}}
==See also==
* [[Lower section]] - the [[dual (order theory)|dual]] concept to this
* [[Uparrow functor|{{M|\Uparrow}}-functor]], a functor, {{M|:}}[[PRE (category)|{{M|\mathrm{PRE} }}]]{{M|\leadsto}}[[TOP (category)|{{M|\mathrm{TOP} }}]] that uses this notion.
** Closely related to the [[Alexandroff topology]]
==References==
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