nLab locally compact locale

Contents

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

Contents

Definition

A locale is locally compact if its frame of opens is a continuous poset.

Properties

Proposition

In classical mathematics, every locally compact locale is spatial, hence may be regarded as a locally compact topological space.

Remark

Prop. may fail in constructive mathematics. For instance any spectrum of a commutative ring, considered as a locale, is locally compact, but may fail to be spatial (because of the lack of filters).

Propsition

Locally compact locales are also exactly the exponentiable objects in the category of locales.

Last revised on June 19, 2022 at 13:24:24. See the history of this page for a list of all contributions to it.