# 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.