# Definition:Category of Open Sets

## 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.