nLab category of open subsets

Definition

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

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

• morphisms are the inclusions

  $\array{ V &&\hookrightarrow && U \\ & \searrow && \swarrow \\ && X }$ of open subsets into each other.

Properties

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

• The category $Op(X)$ 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 $U$:

  $$\bigcup_i U_i = U .$$

  The category of sheaves on $Op(X)$ equipped with this site structure is typically referred to as the category of sheaves on the topological space and denoted

  $$Sh(X) \;\coloneq\; Sh(Op(X)) \,.$$

• The category $Op(X)$ is also a suplattice.

Related concepts

• spatial locale

• spatial topos