nLab locally positive 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 LL is locally positive (or locally (1)(-1)-connected, overt, or open, but not to be confused with an open sublocale) if every open sublocale O𝒪(L)O \in \mathcal{O}(L) is an indexed join of positive open sublocales, or equivalently, if it has a base of positive open sublocales.

Other definitions of a locally positive locale include that the unique map !:L1!:L \to 1 is open, and that the product projection π 2:L×LL\pi_2:L \times L' \to L' is open for every locale LL'.

See also

References

Last revised on September 13, 2024 at 19:29:15. See the history of this page for a list of all contributions to it.