nLab
category of open subsets

Definition

Given a topological space $X$, the category of open subsets $\mathrm{Op}(X)$ of $X$ is the category whose

• objects are the open subsets $U\hookrightarrow X$ of $X$;

• morphisms are the inclusions $\begin{array}{ccccc}V& & \hookrightarrow & & U\\ & \searrow & & \swarrow \\ & & X\end{array}$ of open subsets into each other.

Properties

• The category $\mathrm{Op}(X)$ is a poset, in fact a frame (dually a locale): it is the frame of opens of $X$.

• The category $\mathrm{Op}(X)$ is naturally equipped with the structure of a site, where a collection $\{U_i\to U{\}}_i$ of morphisms is a cover precisely if their union in $X$ equals $U$:

  $$\bigcup _i{U}_i=U.$$\bigcup_i U_i = U .

  The category of sheaves on $\mathrm{Op}(X)$ equipped with this site structure is usually written

  $$\mathrm{Sh}(X):=\mathrm{Sh}(\mathrm{Op}(X)).$$Sh(X) := Sh(Op(X)) \,.