----- > [!proposition] Proposition. ([[the category of preposets is equivalent to that of Alexandrov spaces]]) > The [[category]] $\mathsf{PrePos}$ of [[poset|preordered sets]] and order-preserving maps is [[category equivalence|equivalent]] to the [[category]] $\mathsf{AlexTop}$ of [[Alexandrov topology|Alexandrov]] [[topological space|topological spaces]] and [[continuous|continuous maps]]. Under this equivalence, [[poset|posets]] correspond to [[Alexandrov topology|Alexandrov topologies]] satisfying the $T_{0}$ [[topologically distinguishable|separation axiom]]. > Explicitly, the [[Alexandrov topology|Alexandrov functor]] $\tau:\mathsf{PrePos} \to \mathsf{AlexTop}$ and [[the specialization preorder on a topological space|specialization functor]] $\mathscr{W}:\mathsf{AlexTop} \to \mathsf{PrePos}$ are quasi-inverses. ----- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```