nLab
frame of opens

Context

Topology

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

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

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

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)