Definition:Category of Open Sets

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Definition

Let $T = \struct {S, \tau}$ be a topological space.


Definition 1

The category of open sets of $T$, denoted $\map {\mathbf {Ouv} } T$, is the small category with:

Objects:         open sets of $T$
Morphisms: inclusion mappings between subsets, none otherwise
Composition: composition of mappings
Identity morphisms: identity mappings


Definition 2

The category of open sets of $T$, denoted $\map {\mathbf {Ouv} } T$, is the order category of open sets of $T$ ordered by the subset relation.


Also denoted as

Some authors denote the category $\map {\mathbf {Ouv} } T$ of open sets of $T$ by $\map {\mathbf {Op} } T$ or $\map {\mathbf {Open} } T$.


Also see


Linguistic Note

The notation $\map {\mathbf {Ouv} } T$ for the category of open sets derives from the French ouvert meaning open, in its adjectival form.