# nLab frame of opens

### Context

#### Topology

topology

algebraic topology

topos theory

## Theorems

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 $\Omega \left(X\right)$. 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.

Revised on December 29, 2010 16:08:50 by Urs Schreiber (89.204.137.120)