nLab
category of open subsets

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

Category of open subsets

Definition

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.

Properties

  • 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 usually written

    Sh(X):=Sh(Op(X)). Sh(X) := Sh(Op(X)) \,.

Revised on June 19, 2017 06:22:19 by Urs Schreiber (46.183.103.8)