nLab category of open subsets

Category of open subsets



topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory


Basic concepts

Universal constructions

Extra stuff, structure, properties


Basic statements


Analysis Theorems

topological homotopy theory

Topos Theory

topos theory



Internal Logic

Topos morphisms

Extra stuff, structure, properties

Cohomology and homotopy

In higher category theory


Category of open subsets


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

  • objects are the open subsets UXU \hookrightarrow X of XX;

  • morphisms are the inclusions

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


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

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

    iU i=U. \bigcup_i U_i = U .

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

    Sh(X)∶−Sh(Op(X)). Sh(X) \;\coloneq\; Sh(Op(X)) \,.
  • The category Op(X)Op(X) is also a suplattice.

Last revised on November 5, 2023 at 21:28:11. See the history of this page for a list of all contributions to it.