Definition:Category of Open Sets

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Definition

Let $T = \left({S, \tau}\right)$ be a topological space.


Definition 1

The category of open sets of a topological space $T = \left({S, \tau}\right)$ 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 a topological space $T = \left({S, \tau}\right)$ is the order category of open sets of $T$ ordered by inclusion.


Also see