nLab
Scott topology

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

The Scott topology on a preordered set is the topology in which the open subsets (called Scott-open) are precisely those whose characteristic functions (from the given preorder into the preorder of truth values) preserve directed joins (and this makes them necessarily monotonic).

This in fact ensures that, in general, the continuous functions between preorders with Scott topologies are precisely those (necessarily monotonic) functions between them which preserve directed joins (called Scott-continuous). The poset of truth values itself, therefore, when equipped with the Scott topology, becomes the open-set classifier, Sierpinski space.

Properties

As injective objects in T 0T_0-spaces

In the category of T 0T_0 topological spaces (see at separation axiom), the injective objects are precisely those given by Scott topologies on continuous lattices; as locales these are locally compact and spatial.

Revised on May 1, 2017 03:49:08 by Urs Schreiber (92.218.150.85)