+-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ### Context #### Topology +--{: .hide} [[!include topology - contents]] =-- #### Topos Theory +-- {: .hide} [[!include topos theory - contents]] =-- =-- =-- Given a [[topological space]] $X$, the [[open subspaces]] of $X$ form a [[poset]] which is in fact a [[frame]]. This is the __frame of open subspaces__ of $X$. When thought of as a [[locale]], this is the __[[topological locale]]__ $\Omega(X)$. When thought of as a [[category]], this is the __[[category of open subsets]]__ of $X$. Similarly, given a [[locale]] $X$, the open subspaces of $X$ form a poset which is in fact a frame. This is the __frame of open subspaces__ of $X$. When thought of as a locale, this is simply $X$ all over again. When thought of as a category, this is a [[site]] whose [[topos of sheaves]] is a __[[localic topos]]__. The frame of open subsets of the [[point]] is given by the power set of a singleton, or more generally by the [[subobject classifier|object of truth values]] of the ambient topos. [[!redirects frame of opens]] [[!redirects frames of opens]] [[!redirects frame of open sets]] [[!redirects frames of open sets]] [[!redirects frame of open subsets]] [[!redirects frames of open subsets]] [[!redirects frame of open subspaces]] [[!redirects frames of open subspaces]]