nLab
frame of opens

Context

Topology

Topos Theory

topos theory

Background

Toposes

Internal Logic

Topos morphisms

Extra stuff, structure, properties

Cohomology and homotopy

In higher category theory

Theorems

Given a topological space XX, the open subspaces of XX form a poset which is in fact a frame. This is the frame of open subspaces of XX. When thought of as a locale, this is the topological locale Ω(X)\Omega(X). When thought of as a category, this is the category of open subsets of XX.

Similarly, given a locale XX, the open subspaces of XX form a poset which is in fact a frame. This is the frame of open subspaces of XX. When thought of as a locale, this is simply XX 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 object of truth values of the ambient topos.

Revised on December 30, 2013 11:41:24 by Ingo Blechschmidt (46.244.180.181)