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.