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| In mathematics, a '''partially ordered space''' (or '''pospace''') is a [[topological space]] <math>X</math> equipped with a closed [[partial order]] <math>\leq</math>, i.e. a partial order whose graph <math>\{(x, y) \in X^2 | x \leq y\}</math> is a closed subset of <math>X^2</math>.
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| From pospaces, one can define '''dimaps''', i.e. [[continuous map]]s between pospaces which preserve the order relation.
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| ==Equivalences==
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| For a topological space <math>X</math> equipped with a partial order <math>\leq</math>, the following are equivalent:
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| * <math>X</math> is a partially ordered space.
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| * For all <math>x,y\in X</math> with <math>x \not\leq y</math>, there are open sets <math>U,V\subset X</math> with <math>x\in U, y\in V</math> and <math>u \not\leq v</math> for all <math>u\in U, v\in V</math>.
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| * For all <math>x,y\in X</math> with <math>x \not\leq y</math>, there are disjoint neighbourhoods <math>U</math> of <math>x</math> and <math>V</math> of <math>y</math> such that <math>U</math> is an [[upper set]] and <math>V</math> is a lower set.
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| The [[order topology]] is a special case of this definition, since a [[total order]] is also a partial order. Every pospace is a [[Hausdorff space]]. If we take equality <math>=</math> as the partial order, this definition becomes the definition of a Hausdorff space.
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| == See also ==
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| * [[Ordered vector space]]
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| {{topology-stub}}
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| [[Category:Topological spaces]]
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Latest revision as of 08:24, 30 July 2014
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